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# Nanophotonics

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# Slow light with interleaved p-n junction to enhance performance of integrated Mach-Zehnder silicon modulators

Marco Passoni
/ Dario Gerace
/ Liam O’Faolain
• Centre for Advanced Photonics and Process Analysis, Cork Institute of Technology, Cork, Ireland
• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Lucio Claudio Andreani
Published Online: 2019-05-30 | DOI: https://doi.org/10.1515/nanoph-2019-0045

## Abstract

Slow light is a very important concept in nanophotonics, especially in the context of photonic crystals. In this work, we apply our previous design of band-edge slow light in silicon waveguide gratings [M. Passoni et al, Opt. Express 26, 8470 (2018)] to Mach-Zehnder modulators based on the plasma dispersion effect. The key idea is to employ an interleaved p-n junction with the same periodicity as the grating, in order to achieve optimal matching between the electromagnetic field profile and the depletion regions of the p-n junction. The resulting modulation efficiency is strongly improved as compared to common modulators based on normal rib waveguides, even in a bandwidth of 20–30 nm near the band edge, while the total insertion loss due to free carriers is not increased. The present concept is promising in view of realizing slow-light modulators for silicon photonics with reduced energy dissipation.

## 1 Introduction

Slowing down the propagation of light waves has been a vibrant area of research in recent years [1]. The speed of light can be reduced to values as low as 17 m/s in ultracold atomic gases by the mechanism of electromagnetically induced transparency [2]. Slow light in photonic integrated circuits, especially in silicon photonics, is much less spectacular in terms of slowdown factors, but potentially more interesting for applications due to the enhancement of light-matter interaction. Key issues for practical applications are the group index enhancement, its tradeoff with the bandwidth, and the propagation losses. Much research and efforts have been devoted to the realization of slow light in two-dimensional (2D) photonic crystal slab waveguides [3], [4]. Indeed, photonic crystals (PhCs) possess many degrees of freedom that can be usefully exploited to realize, e.g. flat-band slow light with designed bandwidth and low group-velocity dispersion [5], [6], [7], [8] or the enhancement of nonlinear effects [9], [10], [11]. Nevertheless, photonic crystal waveguides require advanced fabrication and are not employed in industrial devices, which are typically based on conventional silicon ridge/rib waveguides.

Our focus here is the application of slow light to integrated Mach-Zehnder (MZ) modulators in the Silicon-On-Insulator (SOI) platform. MZ modulators for silicon photonics are optical modulators that exploit the variation of the refractive index in one or both arms of a MZ interferometer [12], [13], [14], [15], [16]. Optical modulators are crucial components for optical communication as they are the active elements that encode the information of a bit stream into the optical carrier wave. For high-speed operation required for optical communication, the most appropriate modulation mechanism exploits the plasma dispersion effect in a reverse-biased p-n junction, namely the change of the refractive index due to a variation of the free-carrier concentration [17]. Standard MZ modulators have two arms made of rib waveguides, which include sections with a lateral p-n junction that act as on optical phase-shifter [18], [19]. The main limitation of such a device is the poor overlap between the depletion region of the p-n junction and the optical mode profile. As the resulting change of the effective refractive index is quite small, the modulator length has to be of the order of millimeters, which in turn determines the capacitance and the power dissipation of the device.

In our previous work [20] we optimized the geometry for slow light in waveguide gratings (also known as corrugated or fishbone waveguides) that are made of alternating wide/narrow sections of a rib waveguide in SOI. Waveguide gratings are periodic structures along the direction of propagation that lead to photonic stop bands and to slow light close to the band edges. Such waveguide gratings can be incorporated in standard silicon photonic circuits, as they can be connected with silicon rib waveguides by means of low-loss tapers [21], [22], [23], [24], [25], [26]. In Ref. [20] we designed the optimized parameters of SOI waveguide gratings for a target wavelength in the O-band (λ=1.3 μm) by finding a proper tradoff between the group index enhancement and the slow-light bandwidth. In the present work we optimize phase shifter arms of MZ modulators by matching the optical profile of the slow-light mode with the electrical profile of the free carriers in the p-n junction. This is achieved by using an interleaved p-n junction, which varies periodically along the propagation direction of the optical wave. While previous studies of MZ modulators with either slow light [27], [28], [29] or with interleaved p-n junctions [30], [31], [32], [33] have been reported, the focus here is the synergic use of both concepts. The combination of slow light and interleaved p-n junction is shown here to yield a modulation efficiency that is considerably improved with respect to standard modulators or to the use of both effects alone, while retaining a wide bandwidth that is essential for optical communication. The designed structure fulfills the required constraints for fabrication on industrial technology platforms [34], and may be suitable for the realization of MZ modulators with reduced energy dissipation.

Despite considerable literature on slow light in 2D photonic crystal waveguides, as discussed above, applications of such slow-light structures to electro-optical modulators are uncommon. A review on slow-light MZ modulators based on Si photonic crystals is given in Ref. [35]. We should notice that slow-light MZ modulators with interleaved (or, more generally, modulated) p-n junctions have been deeply studied in the context of 2D PhC waveguides for wavelengths in the C-band (λ=1.55 μm) [36], [37]. The grating waveguides considered here have the advantage of being more compatible with standard silicon photonic circuits on widespread industrial platforms, especially because the structure is simpler and less demanding in terms of optical design and fabrication, and low-loss tapers with standard rib waveguides are easily available. Resonator-based modulators with interleaved p-i-n junctions leading to matching with optical mode have also been developed [38], [39], [40]. Such resonators have typically a small bandwidth, unlike the MZ modulators proposed in this work.

The rest of this paper is organized as follows: in Section 2 we briefly review the optimization of band-edge slow-light structures. In Section 3 we describe the interleaved p-n junction and the corresponding electrical simulations. In Section 4 we present the results of electro-optical simulations, which are applied in Section 5 to calculate the modulation efficiency and the propagation losses of phase shifters. The dynamic behavior is also calculated. Section 6 contains a summary and discussion of the results.

## 2 Slow light in silicon waveguide gratings

The waveguide gratings considered in this work are displayed in Figure 1, and consist of a silicon rib waveguide with periodically modulated waveguide width embedded in SiO2. The slow-light behavior arises from the flattening of the band dispersion at the edge of a photonic band gap. These structures were systematically studied in our previous work [20], and two optimized configurations will be analyzed here. The common parameters of both structures are the waveguide thickness s=310 nm, the fill-factor d1/a=0.5 where a=d1+d2 is the period, and the widths W1 and W2 that are equal to 100 nm and 800 nm, respectively. The two configurations differ mainly for the silicon thickness in the etched cladding regions, t: the first structure has t=150 nm and is named SLOW150, while the second structure has t=50 nm and is denoted as SLOW50. These values correspond to common technology platforms for optical interconnects [34]. To lock the lowest band edge at λ=1.3 μm, the period has to be fixed as a=218.1 nm and a=234.7 nm for SLOW150 and SLOW50 configurations, respectively.

Figure 1:

Structure of the slow-light waveguide: 3D view (upper panel) and cross sections (planes A, B, C) with parameters definition.

Brown is Si, while the surrounding medium is SiO2.

The photonic band dispersion and group index of the lowest TE mode are plotted in Figure 2 for both SLOW150 and SLOW50 configurations. Both Si and SiO2 are treated as dispersionless materials with a dielectric constant ε=12.299 and 2.093, respectively. The increase of the group index in the proximity of the band edge is clearly evident, and it is more prominent in the SLOW50 case, due to the greater strength of the periodic modulation in this configuration. We see, for example, that a group index ng>10 is compatible with a bandwidth of ~10 nm. Notice that the photonic dispersion lies entirely below the air light line, thus the lowest TE mode is truly guided with no intrinsic losses, while actual propagation losses will be of extrinsic type and related to disorder-induced scattering at the vertical sidewall roughness [41], [42]. The designs fulfill the requirement of a minimum feature size ~100 nm, thus the structures may be fabricated by deep UV lithography with low losses.

Figure 2:

Fundamental band dispersion and group index of the two slow-light configurations adopted in this work.

## 3 Electrical structure and simulation

In order to enhance the efficiency of a phase-shifter based on the plasma dispersion effect, it is very important to maximize the overlap between the region in which the refractive index is modified – i.e. the voltage-dependent depletion region of the p-n junction – and the optical mode of the waveguide. In phase shifters based on a conventional rib waveguide, this is usually achieved by using a lateral p-n junction with an optimized junction position [43]. More complex geometries like interleaved [33] or U-shaped [44] p-n junctions have also been considered. In these cases, the geometry of the p-n junction is chosen in order to optimize the overlap with the optical mode in the transverse directions, i.e. perpendicular to the waveguide axis.

The scheme proposed in this work consists in an interleaved p-n junction whose period is twice the period of the waveguide grating and with a trench depth Wi of 1 μm, see Figure 3A and B. The goal of such a structure is not only to increase the overlap between the depletion region and the optical mode in the transverse direction, but also to exploit the reshaping of the field profile caused by the slow-light waveguide. Indeed, the electric field – see Figure 3C – has the typical profile of an optical mode near the low energy edge of a band gap, i.e. it is a dielectric mode featuring a strong localization of the field in the regions with higher (effective) index, i.e. in the thicker sections of the grating. As a consequence, this particular doping arrangement provides an ideal superposition between the depletion regions of the junction and the dielectric regions in which the electromagnetic field is more intense.

Figure 3:

Doping distribution: 3D view (A) and top view (B). In addition, field profile of the optical mode (C) for the SLOW50 waveguide at λ=1.32 μm.

We assume doping concentrations to be N=1.4×1018 cm−3 and P=5×1017 cm−3 as in the literature [43]. Then, we calculate the charge density in the junction as a function of the applied reverse voltage by employing the software Lumerical DEVICE. The code solves the drift-diffusion equations, by applying the finite element method over an unstructured grid [45]. Due to the periodic nature of the structure, the analysis of only one period a is needed, and symmetric boundary conditions are applied along the waveguide direction. The obtained distribution is then interpolated on a regular grid, for clearer visualization and further processing.

As we can see from the example in Figure 4, the depletion region lies in the center of the wider section of the waveguide grating and its width increases with the applied voltage. As a consequence, the refractive index of the waveguide is modified by the plasma dispersion effect. This leads to a phase shift of the propagating optical mode, which is calculated in the next Section. While the charge profiles of Figure 4 are representative of the general behaviour of the interleaved p-n junction in the grating waveguide, the detailed features of the depletion region depend on the exact values of the dopings and also on a possible offset between the junction position and the center of the wide sections. Using a smaller p-type than n-type doping is justified by the fact that the plasma-dispersion effect [Eq. (1) below] is stronger for holes than for electrons. A full optimization of doping levels and profiles has to consider the tradeoffs among various figures of merit (modulator efficiency, losses, cut-off frequency…) and is left for future work. We should also notice that possible deviations or inhomogeneities in doping levels between the unetched and etched regions of the silicon material (as pointed out in Ref. [46]) would not affect the main feature of the present structure, namely the same periodicity of electrical and optical profiles.

Figure 4:

Doping profile within one single period (A) and cross-section of charge distribution as a function of voltage (B–F) for the SLOW50 waveguide. The snapshots are taken at a height of 150 nm from the bottom of the waveguide.

## 4 Optical simulation

Once the distribution of the free carriers is known, the effect of the applied voltage on the optical mode propagation can be quantified. For this purpose, we first describe the plasma-dispersion effect by converting the charge distribution into a dielectric constant variation using Soref-like equations [17] with textbook coefficients [18]:

$Δn=−3.64⋅10−10λ2N−3.51⋅10−6λ2P0.8,$(1)

$Δk=2.80⋅10−5λ3N+1.91⋅10−5λ3P,$(2)

where N, P are the electrons and holes densities in cm−3, λ is the wavelength in meters (here λ=1.3 μm), Δn and Δk are the differences in the real and imaginary part of the refractive index, respectively. The resulting dielectric constant εε, where $\Delta \epsilon =2\sqrt{\epsilon }\left(\Delta n+i\Delta k\right),$ is plugged into the optical simulation, which then calculates the photonic mode dispersion by the aperiodic Fourier-modal method (A-FMM) [47], [20]. In the process, the dielectric constant in each unit cell is discretized using 40 slices, thus with a spatial resolution of about 5 nm. The required Fourier transforms within a single slice are handled numerically.

In order to describe the performance of the phase shifter in terms of modulation efficiency and propagation losses, we need to calculate the photonic mode dispersion in a periodic grating. In practice, two kinds of optical simulations can be performed. When only the change in the dispersion is needed (i.e. for calculating the phase shift), the optical simulation can be done considering only the change in the real part of the dielectric constant. This greatly simplifies the treatment, since the Bloch vector is purely real and the fitting procedure described in the Appendix can be used. The more complete simulation, including also the imaginary part of the dielectric constant, is used to calculate the propagation loss due to free-carrier absorption, which can be extracted from the imaginary part of the Bloch vector.

In both cases, we need to calculate the scattering matrix for one period of the structure. Due to the plasma dispersion effect, the period 2a of the waveguide with interleaved doping is twice the period a of the waveguide grating, see Figure 3A and B. However, a simplification can be made by noting that adjacent periods of the waveguide grating, which are different from the electrical point of view, are the mirror image of each other and produce the same phase-shift for the optical mode. We therefore treat two adjacent periods as being identical in the optical simulation, which means that the doped waveguide is still treated as a periodic medium with period a. This assumption greatly simplifies the calculation, as the resulting band dispersion for the structure with period a can be fitted with an analytic formula with few parameters, as explained in the Appendix. We checked the validity of this approximation by comparing the resulting photonic dispersion with that calculated for the full doped waveguide with period 2a, finding excellent agreement (deviations on the band dispersion <10−7, compared to ~10−3 variation due to the applied voltage).

The effect of the applied voltage on the band dispersion is displayed in Figure 5 as the difference in the effective (phase) index of the optical modes between the biased and unbiased waveguide. The enhancement due to slow light is evident, as shown by the increase of the effective index difference towards the band edge. As expected, this effect is stronger in the SLOW50 than in the SLOW150 structure, since the SLOW50 structure has an increased slowdown factor that enhances the plasma dispersion effect.

Figure 5:

Difference in effective (phase) index between the biased and un-biased waveguides at different values of applied reverse voltage for SLOW150 (A) and SLOW50 (B) configurations.

In addition, in Figure 6 we report the excess loss due to free-carrier absorption for both slow-light configurations, extracted from the imaginary part of the effective index of the mode as α=(20/ln 10)·ωkeff/c. The plotted results refer to the unbiased waveguides, and represent in a sense the worst-case scenario, since reverse-biasing the junction reduces the absorption loss by lowering the free-carrier density in the waveguide. It can be seen that both configurations exhibit an increased propagation loss towards the band edge, as is well known to happen for slow light. However, the relevant quantity is the total insertion loss of the phase shifter, which is evaluated in the next Section.

Figure 6:

Additional loss due to free-carrier absorption for both SLOW150 and SLOW50 configurations.

## 5 Results on phase-shifters

The main figure of merit for a phase-shifter is usually taken to be VπLπ, namely, the product of the driving voltage and of the length that is needed to yield a π phase difference with respect to the unbiased case. VπLπ is commonly referred to as the modulation efficiency, and it generally increases as a function of the applied reverse voltage V, since the width of the depletion region increases sublinearly with V and so does the phase shift at a fixed length. In general, VπLπ depends also on the wavelength, but in phase-shifters based on conventional rib waveguides this dependence is negligible. This is not the case in our slow-light waveguides, because the phase modulation is closely related to the group index, which depends strongly on the wavelength.

As for the insertion loss, a common figure of merit is IL(Lπ), obtained as the propagation loss per unit length α times the length Lπ. Here, α is calculated at V=0, while Lπ depends on the voltage: this should again be viewed as a quantification of losses in a worst-case scenario, and it applies to MZ modulators in a push-pull configuration where one of the two phase shifters is always kept at V=0 [19]. A typical modulator operates with a phase shift ~0.15π in each arm, thus the length has to be L=0.15Lπ and the actual insertion loss is ~0.15·IL(Lπ).

The obtained VπLπ as a function of wavelength is reported in Figure 7 for both SLOW150 and SLOW50 configurations. The effect of slow light is clearly evident, as VπLπ decreases monotonically when moving towards the band edge, where the group index increases. Taking advantage of this effect, values of VπLπ around 0.1 V cm can be reached, although within a small spectral window (1-2 nm wide) in the proximity of the band edge. This value for the modulation efficiency is about one order of magnitude lower than those obtained in conventional ridge waveguides, where VπLπ is typically above 1 V cm [15]. Performance of the slow-light structures remains good even when a greater bandwidth is required. In particular, the SLOW50 configuration at low voltage shows a VπLπ lower than 0.3 V cm over a spectral window of about 30 nm, which is adequate for many telecom applications. This wide bandwidth follows from the spatial matching of the electromagnetic field profile with the depletion region of the p-n junction, which is maintained in a wide spectral region below the band edge.

Figure 7:

Modulation efficiency as a function of wavelength and applied reverse bias for SLOW150 (A) and SLOW50 (B) configurations.

To see the advantage of the present slow-light design, we present in Table 1 a comparison between the calculated performance of the SLOW50 structure and those of two representative rib waveguides with lateral and interleaved p-n junctions, respectively [33]. Considering first the modulation efficiency VπLπ, we notice that the interleaved p-n junction gives an improvement over the lateral p-n junction even in a normal rib waveguide configuration [33]. When the slow-light structure is considered, VπLπ is further reduced to values as low as 0.1–0.3 V cm at 5–10 nm from the band edge λ=1.3 μm. This clearly shows the advantage of combining slow light with the interleaved p-n junction, which strongly improves the performance in a wide bandwidth. Considering now the losses, the propagation loss in dB/cm increases, when going from the normal, to the interleaved, to the slow-light structures: however, IL(Lπ) is nearly the same for all configurations. In practice, this means that the optimal length of a slow-light modulator can be a fraction of a mm, i.e. much smaller than in conventional modulators, thus yielding the same total insertion loss but with a strongly improved modulation efficiency.

Table 1:

Modulation efficiency VπLπ, loss per unit length at V=0, and insertion loss IL(Lπ). Both VπLπ and IL(Lπ) are given for two different values of the applied reverse bias.

Reducing the modulator length has also the advantage of improving tolerance against disorder-induced losses due to fabrication processes. A modulator length below 500 μm means that propagation losses up to several 10 dB/cm can be tolerated, which is well within the reach of current fabrication technologies, with margins of improvement thanks to the use of immersion lithography [16].

It is also interesting to quantify the effect of p-n junction geometry in the slow-light structure. For this purpose, we calculate the figures of merit for the SLOW50 configuration but using a lateral p-n junction with the same doping levels. At λ=1330, 1320, 1310, 1305 nm and V=1 V we obtain VπLπ=0.94, 0.79, 0.59, 0.45 V cm, respectively, which is nearly three times larger than for the interleaved p-n junction. For the same wavelengths, the propagation losses at V=0 are 28.7, 34.0, 45.8, 60.4 dB/cm, leading to IL(Lπ) ~27 dB at 1 V, which is much higher than the values reported in Table 1. Thus, we conclude that the use of the interleaved p-n junction greatly improves the efficiency and reduces the insertion loss of the slow-light modulator, thereby demonstrating the key role of spatial matching between the p-n junction and the optical field profile.

To calculate the frequency response of the phase shifter, we model the structure as an RC circuit and analyse the small-signal response with Lumerical DEVICE. The resistance and the capacitance are obtained from the complex impedance Z=R+1/(jωC). The 3 dB cutoff frequency is calculated as f3dB=1/(2πRC), and it is independent of the modulator length L, since CL and R∝1/L. The simulation setup now includes additional highly doped p+ and n+ regions, see Figure 8A. We choose Wd=1.2 μm, a total width Wt=2.0 μm, and dopings of 1×1019 cm−3 in the p+ and n+ regions, with metal contacts at the borders. With these values the waveguide propagation losses are not increased over those reported in Figure 6, as we have verified. The results as a function of bias voltage are reported in Figure 8. The behavior is similar for the two slow-light configurations, showing an increase of resistance and a decrease of capacitance for increasing voltage, both effects being caused by an expansion of the depletion region. The SLOW50 configuration has higher resistance and lower capacitance compared to the SLOW150 configuration, the first effect dominates and leads to a smaller cut-off frequency. The SLOW150 configuration can reach f3dB>12.5 GHz, which can sustain a bit rate up to 25 Gbps with non-return to zero format, or even up to 50 Gbps with PAM-4 encoding [48]. The corresponding rates for the SLOW50 configuration are about a factor of two smaller. There is much room for improving the dynamic behavior by tailoring the doping levels and the spatial profiles of both low- and highly doped regions. Systematic optimization is left for future work.

Figure 8:

(A) Structure layout for calculating the cutoff frequency, (B) resistance times length, (C) capacitance per unit length, (D) 3-dB cutoff frequency f3dB.

The advantage of the slow-light modulator design becomes more useful at small driving voltage, which has a beneficial effect on the energy dissipation given by CV2/2. To give an example, we consider both SLOW50 and SLOW150 configurations at 1 V reverse bias. The capacitances per unit length (considering the average values at 0.5 V) are C/L=1.8 pF/mm and 2 pF/mm, respectively. Working at λ=1310 nm, the efficiencies are VπLπ=0.191 V cm and 0.297 V cm, therefore a phase shift of 0.15π is obtained with length L=0.15 Lπ=290 μm for SLOW50 and L=450 μm for SLOW150 configuration. The energy dissipation is then 0.26 pJ/bit for SLOW50 and 0.45 pJ/bit for SLOW150. These values are reduced up to one order of magnitude compared to typical modulators based on normal rib waveguides [14], [19], and are comparable to those obtained with PhC waveguides [37].

## 6 Conclusion

Summarizing, we propose a new strategy for exploiting slow light in silicon Mach-Zehnder modulators. The idea is to combine band-edge slow light in a waveguide grating with an interleaved p-n junction along the propagation direction, which has the same period of the optical waveguide. This leads to optimal matching between the propagating electric field and the depletion regions of the p-n junctions, and results in a strongly improved modulation efficiency over a bandwidth up to 20–30 nm. The use of an interleaved p-n junction is especially useful to increase the bandwidth of the slow-light modulator, as the spatial matching between the depletion region and the field profile in the waveguide grating is wavelength-independent. The modulation efficiency VπLπ is in the range 0.1–0.5 V cm, depending on the operating bandwidth and on the driving voltage, the best performances being obtained at low voltage V~1 V.

It is important to notice that the total loss of the slow-light design presented here is not increased with respect to modulators based on normal rib waveguides. Interestingly, the use of an interleaved p-n junction significantly reduces the loss of the slow-light structure relative to a lateral p-n junction: e.g., IL(Lπ) ~27 dB and ~6.5 dB with lateral and interleaved p-n junction, respectively (the corresponding insertion losses for a length L=0.15Lπ are 4 and 1 dB). In summary, the use of the interleaved p-n junction in the slow-light modulator improves the efficiency and reduces the free-carrier induced insertion losses.

A notable simplification in these time-consuming calculations is given by the procedure that relies on the photonic band structure for a single period, which is fitted with a simple analytic formula. It should be remarked that the present design is not especially optimized. Further improvements of the modulation efficiency are expected by optimizing the doping and/or the junction position inside the waveguide. The present structures can support high-speed modulation, and there is room for improving the cutoff frequency by tuning the doping levels and doping profiles both inside and outside the waveguide. Clearly, a proper tradeoff between modulation rate, losses, and energy consumption will have to be determined for specific applications. Thus, we expect that further improvement of modulator performance will be possible, and that the present concepts can find application to reduce the energy dissipation in MZ modulators on current-day technology platforms.

## A Phase shift, fitting procedure and parameters

Once the photonic band dispersion in the waveguide grating is calculated, the phase shift of one modulator arm is obtained as Δϕ(ω)=k(ω)L, where k(ω) is the wavevector as a function of frequency and L is the length of the phase shifter arm. The calculations are greatly simplified by fitting the band dispersion with an analytic formula that results from perturbation theory [20]. The dimensionless frequency $\Omega =\frac{\omega a}{2\pi c}$ is related to the dimensionless distance from the band edge $\delta =\frac{\pi }{a}\left(1-k\right)$ using the following formula:

$Ω(δ)=[Ω02+δ2−4δ2+(1−δ2)2U24n2]12,$(3)

which is the corrected form of the one reported in [20]. The fitting parameters Ω0, n, U represent the midgap frequency, the effective index, and the coupling factor, respectively.

Since the dispersions of the same waveguide at different voltages are quite close to each other, a great precision is needed in the fit. For this reason, in this work the fitting procedure is carried out using up to 100 energy points for each curve. This allows to capture even the small differences in the dispersion as a function of the applied voltage.

Once the fitting parameters and the period are known, it is thus possible to use Eq. (3) to reconstruct the relation between frequency ω and wavevector k. The modulation efficiency is then obtained as VπLπ=V·πk(ω), where Δk=kV)−k0(ω) is the change in wavevector induced by the plasma dispersion effect. For completeness we report in Table 2, the fitting parameters, as a function of reverse bias, for both SLOW150 and SLOW50 configurations.

Table 2:

Fitting parameters for both slow-light configurations at different voltages.

## Acknowledgments

The authors are grateful to C. Baudot, A. Fincato, A. Ghilioni, R. Sheehan, A. Simbula, F. Svelto, E. Temporiti for many useful discussions and suggestions. We acknowledge the CINECA-ISCRA award for the availability of high performance computing resources and support. This work was supported by the European Union under H2020-ICT27-2015 project no. 688516 COSMICC, Funder Id: http://dx.doi.org/10.13039/100010661, and by EU H2020 QuantERA ERA-NET Co-fund in Quantum Technologies project CUSPIDOR, Funder Id: http://dx.doi.org/10.13039/501100003407, co-funded by Italian MIUR and Science Foundation Ireland under grant 17/QERA/3472.

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## About the article

Received: 2019-02-15

Revised: 2019-04-12

Accepted: 2019-04-13

Published Online: 2019-05-30

Citation Information: Nanophotonics, Volume 8, Issue 9, Pages 1485–1494, ISSN (Online) 2192-8614,

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© 2019 Lucio Claudio Andreani et al., published by De Gruyter, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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## Mass generation: The solution

26/12/2010

In my preceding post I have pointed out an interesting mathematicalquestion about the exact solutions of the scalar field theory that I use in this paper

$\Box\phi+\lambda\phi^3=0$

given by

$\phi=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\cdot x,i)$

that holds for

$p^2=\mu^2\sqrt{\frac{\lambda}{2}}.$

If you compute the Hamiltonian the energy does not appear to be finite, differently from what the relation dispersion is saying. This is very similar to what happens to plane waves for the wave equation. The way out is to take a finite volume and normalize properly the plane waves. One does this to get the integral of the Hamiltonian finite and all amounts to a proper normalization. In our case where must this normalization enter? The striking answer is: The coupling. This is an arbitrary parameter of the theory and we can properly rescale it to get the right normalization in the Hamiltonian. The final result is a running coupling exactly in the same way as I and others have obtained for the quantum theory. You can see the coupling entering in the right way both in the solution and in the computation of the Hamiltonian.

If you are curious about these computations you can read the revised version of my paper to appear soon on arxiv.

Marco Frasca (2010). Mass generation and supersymmetry arxiv arXiv: 1007.5275v1

format: plain


text:
Ask question

# At a certain university 523 of the seniors are history majors or math majors(or both).There are 100 senior math majors, and 33 seniors are majoring in both history and math. How many seniors are majoring in history?

Question
Upper Level Math
asked 2020-12-16
At a certain university 523 of the seniors are history majors or math majors(or both).There are 100 senior math majors, and 33 seniors are majoring in both history and math. How many seniors are majoring in history?

## Answers (1)

2020-12-17
Since you have asked multiple questions, we will solve the first question for you. If you want any specific question to be solved then please specify the question number or post only that question.
1)
Inclusion-Exclusion Formula:
If A and B are any two finite sets, then $$\displaystyle{\left|{{A}\cup{B}}\right|}={\left|{{A}}\right|}+{\left|{{B}}\right|}-{\left|{{A}\cap{B}}\right|}$$.
Let H, M be the sets of seniors whose major is History and Math respectively.
It is given that 523 seniors are history majors or math majors or both.
Thus, $$\displaystyle{\left|{{H}\cup{M}}\right|}={523}$$.
It is given that 100 seniors are math majors.
Thus, $$\displaystyle{\left|{{M}}\right|}={100}$$.
The number of seniors majoring in math and history are 33.
Thus, $$\displaystyle{\left|{{H}\cap{M}}\right|}={33}$$.
Step 2
Evaluate the number of seniors majoring in History using Inclusion-Exclusion principle as follows.
$$\displaystyle{\left|{{H}\cup{M}}\right|}={\left|{{H}}\right|}+{\left|{{M}}\right|}-{\left|{{H}\cap{M}}\right|}$$
$$\displaystyle{523}={\left|{{H}}\right|}+{100}-{33}$$
$$\displaystyle{523}={\left|{{H}}\right|}+{67}$$
$$\displaystyle{\left|{{H}}\right|}={523}-{67}$$
$$\displaystyle{\left|{{H}}\right|}={456}$$
Therefore, the number of seniors majoring in History is 456.

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...


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# Why does chocolate ice cream melt faster than others?

The other day I was dipping out Neapolitan(strawberry, vanilla, & chocolate) ice cream and noticed that the chocolate ice cream tended to melt faster than that of the vanilla and strawberry. I couldn't remember in which order I dipped the ice cream so I decided to do a controlled experiment with it the next time I dipped it out.

I took three bowls of the same type/shape and dipped out the strawberry first, then vanilla, and chocolate last. Even dipping the chocolate ice cream last showed that the chocolate ice cream would melt faster than the strawberry or vanilla.

Neither the strawberry nor chocolate use artificial coloring, though the vanilla states that it does. Is there a reason as to why chocolate ice cream melts faster? Is it because of the darker coloring that attracts heat or perhaps the low melting temperature of chocolate in general?

• The darker color hypothesis sounds reasonable, how can you change your experiment to test this hypothesis? – Leonardo Dec 3 '12 at 7:20
• what about different melting points? – mdpc Dec 3 '12 at 20:05
• @mdpc Could you explain, in your answer, why you think they have different melting points? Please include some chemistry concepts in your answer.. (I have converted to comment for now) – ManishEarth Dec 4 '12 at 2:50

## 2 Answers

It's a complicated problem and I'm going to add a hand-waving physics answer.

For most materials, at a given wavelength the albedo and and emissivity sum to roughly unity. So if something is close to white with an albedo of 0.9, it's emissivity in the visible range is about 0.1, and if something is dark, those are reversed with albedo ~0.1 and emissivity ~0.9.

However, in the thermal infrared many non-metallic materials have an emissivity of very roughly 0.9 no matter what color they are in the visible. Even glass and water which are transparent in the visible have a high infrared emissivity.

In direct sunshine a surface perpendicular to the light will receive about 1100 W/m^2 in the visible, and so chocolate ice cream will be heated much faster than vanilla. Strawberry is pink because it absorbs shorter wavelengths but there's not a lot of power there from the Sun's black-body spectrum, so strawberry is like vanilla in the Sun.

However on an overcast day, or indoors, the the light is roughly 50 to 100 time dimmer, or about 11 to 22 W/m^2

(You can prove that by shooting a photo at f/16 with a shutter speed of 1/ISO or 1/ASA. I'm dating myself, but if had ASA 64 film you'd shoot f/16 at 1/60th second outdoors. You'd get nothing if you left it that way and shot indoors (Sunny 16 rule)

So now we're only down to say 11 or 22 W/m^2 in the visible, but that's not all the light! The Stefan-Boltzmann law tells us

$$P = \sigma \epsilon T^4.$$

If you plug in 273K and 298K (0° and 25° C) and $$\epsilon=0.9$$, you get about 280 and 400 W/m^2. That means that you, the walls ceiling or clouds are heating your ice cream at 400 W/m^2 while it is only radiating 280 W/m^2 back, it's a difference of 100 W/m^2 or way more than the heating at visible wavelengths.

So I would propose that if the phenomenon happens indoors that it's not related to the color, and could be related to the recipe.

Maybe chocolate ice cream has more salt, depressing the melting point, and so it starts melting sooner because it melts at a lower temperature.

This appears to be physics more than chemistry. A dark color means it emits less visible light but rather absorbes it with temperature rise. Have your next ice cream in a darker place as to visible and ir emisions to test this.

• This is almost certainly a factor, but anecdotally I have also observed that chocolate ice cream is softer, even straight from the freezer. This would likely be due to some change in the mixture of substances in chocolate ice cream versus other flavors. I wouldn't go so far as to say it's any one thing, but it's a different recipe so it will have different properties. – KeithS Jul 23 '13 at 22:41


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RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Differ. Uravn.: Year: Volume: Issue: Page: Find

 Differ. Uravn., 1996, Volume 32, Number 3, Pages 357–374 (Mi de8955)

Partial Differential Equations

An order exact estimate uniform in $\mathbf R^N$ for $N=2$ and $N=3$, for the squares of fundamental functions of a selfadjoint extension in $\mathbf R^N$ of the Schrödinger operator with a potential satisfying Kato's condition

V. A. Il'ina, E. I. Moiseevb

a Lomonosov Moscow State University
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Full text: PDF file (2502 kB)

English version:
Differential Equations, 1996, 32:3, 364–379

Bibliographic databases:
UDC: 517.984.5

Citation: V. A. Il'in, E. I. Moiseev, “An order exact estimate uniform in $\mathbf R^N$ for $N=2$ and $N=3$, for the squares of fundamental functions of a selfadjoint extension in $\mathbf R^N$ of the Schrödinger operator with a potential satisfying Kato's condition”, Differ. Uravn., 32:3 (1996), 357–374; Differ. Equ., 32:3 (1996), 364–379

Citation in format AMSBIB
\Bibitem{IliMoi96} \by V.~A.~Il'in, E.~I.~Moiseev \paper An order exact estimate uniform in $\mathbf R^N$ for $N=2$ and $N=3$, for the squares of fundamental functions of a selfadjoint extension in $\mathbf R^N$ of the Schr\"odinger operator with a potential satisfying Kato's condition \jour Differ. Uravn. \yr 1996 \vol 32 \issue 3 \pages 357--374 \mathnet{http://mi.mathnet.ru/de8955} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1435114} \transl \jour Differ. Equ. \yr 1996 \vol 32 \issue 3 \pages 364--379


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• ### Workshop on Physics with Neutral Kaon Beam at JLab (KL2016) Mini-Proceedings(1604.02141)

April 6, 2016 hep-ph, hep-ex, nucl-ex, nucl-th
The KL2016 Workshop is following the Letter of Intent LoI12-15-001 "Physics Opportunities with Secondary KL beam at JLab" submitted to PAC43 with the main focus on the physics of excited hyperons produced by the Kaon beam on unpolarized and polarized targets with GlueX setup in Hall D. Such studies will broaden a physics program of hadron spectroscopy extending it to the strange sector. The Workshop was organized to get a feedback from the community to strengthen physics motivation of the LoI and prepare a full proposal. Further details about the Workshop can be found on the web page of the conference: http://www.jlab.org/conferences/kl2016/index.html .
• ### A study of decays to strange final states with GlueX in Hall D using components of the BaBar DIRC(1408.0215)

Aug. 1, 2014 hep-ex, physics.ins-det
We propose to enhance the kaon identification capabilities of the GlueX detector by constructing an FDIRC (Focusing Detection of Internally Reflected Cherenkov) detector utilizing the decommissioned BaBar DIRC components. The GlueX FDIRC would significantly enhance the GlueX physics program by allowing one to search for and study hybrid mesons decaying into kaon final states. Such systematic studies of kaon final states are essential for inferring the quark flavor content of hybrid and conventional mesons. The GlueX FDIRC would reuse one-third of the synthetic fused silica bars that were utilized in the BaBar DIRC. A new focussing photon camera, read out with large area photodetectors, would be developed. We propose operating the enhanced GlueX detector in Hall D for a total of 220 days at an average intensity of 5x10^7 {\gamma}/s, a program that was conditionally approved by PAC39
• ### An initial study of mesons and baryons containing strange quarks with GlueX(1305.1523)

May 7, 2013 nucl-ex
The primary motivation of the GlueX experiment is to search for and ultimately study the pattern of gluonic excitations in the meson spectrum produced in $\gamma p$ collisions. Recent lattice QCD calculations predict a rich spectrum of hybrid mesons that have both exotic and non-exotic $J^{PC}$, corresponding to $q\bar{q}$ states ($q=u,$ $d,$ or $s$) coupled with a gluonic field. A thorough study of the hybrid spectrum, including the identification of the isovector triplet, with charges 0 and $\pm1$, and both isoscalar members, $|s\bar{s}\ >$ and $|u\bar{u}\ > + |d\bar{d}\ >$, for each predicted hybrid combination of $J^{PC}$, may only be achieved by conducting a systematic amplitude analysis of many different hadronic final states. Detailed studies of the performance of the \gx detector have indicated that identification of particular final states with kaons is possible using the baseline detector configuration. The efficiency of kaon detection coupled with the relatively lower production cross section for particles containing hidden strangeness will require a high intensity run in order for analyses of such states to be feasible. We propose to collect a total of 200 days of physics analysis data at an average intensity of $5\times 10^7$ tagged photons on target per second. This data sample will provide an order of magnitude statistical improvement over the initial GlueX running, which will allow us to begin a program of studying mesons and baryons containing strange quarks. In addition, the increased intensity will permit us to study reactions that may have been statistically limited in the initial phases of GlueX. Overall, this will lead to a significant increase in the potential for \gx to make key experimental advances in our knowledge of hybrid mesons and excited $\Xi$ baryons.
• ### A study of meson and baryon decays to strange final states with GlueX in Hall D (A proposal to the 39th Jefferson Lab Program Advisory Committee)(1210.4508)

Oct. 16, 2012 hep-ex, nucl-ex
The primary motivation of the GlueX experiment is to search for and ultimately study the pattern of gluonic excitations in the meson spectrum produced in gamma p collisions. Recent lattice QCD calculations predict a rich spectrum of hybrid mesons that have both exotic and non-exotic JPC, corresponding to q q-bar (q=u, d, or s) states coupled with a gluonic field. A thorough study of the hybrid spectrum, including the identification of the isovector triplet, with charges 0 and +-1, and both isoscalar members, |s s-bar> and |u u-bar> + |d d-bar>, for each predicted hybrid combination of JPC, may only be achieved by conducting a systematic amplitude analysis of many different hadronic final states. We propose the development of a kaon identification system, supplementing the existing GlueX forward time-of-flight detector, in order to cleanly select meson and baryon decay channels that include kaons. Once this detector has been installed and commissioned, we plan to collect a total of 200 days of physics analysis data at an average intensity of 5 x 10^7 tagged photons on target per second. This data sample will provide an order of magnitude statistical improvement over the initial GlueX data set and, with the developed kaon identification system, a significant increase in the potential for GlueX to make key experimental advances in our knowledge of hybrid mesons and Cascade baryons.


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## The partition of unity finite element method: basic theory and applications.(English)Zbl 0881.65099

Summary: The paper presents the basic ideas and the mathematical foundation of the partition of unity finite element method (PUFEM). We will show how the PUFEM can be used to employ the structure of the differential equation under consideration to construct effective and robust methods. Although the method and its theory are valid in $$n$$ dimensions, a detailed and illustrative analysis will be given for a one-dimensional model problem. We identify some classes of non-standard problems which can profit highly from the advantages of the PUFEM and conclude this paper with some open questions concerning implementational aspects of the PUFEM.

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
Full Text:

### References:

 [1] () [2] I. Babuška and J.E. Osborn, Private communication. [3] Bergman, S., Integral operators in the theory of linear partial differential equations, (1961), Springer Verlag · Zbl 0093.28701 [4] Freund, R.W., A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems, SIAM J. sci. comput., 14, 2, 470-482, (1993) · Zbl 0781.65022 [5] Gradshtein, I.S., Table of integrals, series and products, (1980), Academic Press New York · Zbl 0102.24703 [6] Herrera, I., Boundary methods: an algebraic theory, (1984), Pitman Boston · Zbl 0549.35004 [7] Kress, R., Linear integral equations, (1989), Springer Verlag [8] Melenk, J.M., On generalized finite element methods, () · Zbl 0941.65112 [9] Mergelyan, S.N., (), 294-391 [10] Muskhelishvili, N.I., Some basic problems of the mathematical theory of elasticity, (1963), P. Noordhoff Groningen · Zbl 0124.17404 [11] Oh, Hae-Soo; Babuška, I., The p-version of the finite element method for the elliptic boundary value problems with interfaces, Comput. methods appl. mech. engrg., 97, 211-231, (1992) · Zbl 0762.65059 [12] Oh, Hae-Soo; Babuška, I., The method of auxiliary mapping for the finite element solutions of elasticity problems containing singularities, J. comput. phys., 121, 2, 193-212, (1995) · Zbl 0833.73061 [13] Babuška, I.; Caloz, G.; Osborn, J., Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. numer. anal., 31, 945-981, (1994) · Zbl 0807.65114 [14] Babuška, I.; Ihlenburg, F.; Paik, E.; Sauter, S., A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution, Comput. methods appl. mech. engrg., 128, 325-360, (1995) · Zbl 0863.73055 [15] I. Babuška and J.M. Melenk, The partition of unity method, Int. J. Numer. Methods Engrg., in press. [16] I. Babuška and Z. Zhang, The partition of unity finite element method for the elastically supported beam, to appear. [17] Szegö, G., Über polynome, die zu einer gegebenen kurve der komplexen ebene gehören, Mathematische zeitschrift, 9, 218-270, (1921) · JFM 48.0374.04 [18] Thomson, L.L.; Pinsky, P.M., A Galerkin least squares finite element method for the two-dimensional Helmholtz equation, Int. J. numer. methods engrg., 38, 371-397, (1995) · Zbl 0844.76060 [19] Vekua, I.N., New methods for solving elliptic equations, (1967), North Holland · Zbl 0146.34301 [20] Walsh, J.L., Interpolation and approximation by rational functions in the complex domain, () · Zbl 0013.05903
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## anonymous 3 years ago If f(x) = integral (3,x) sqr root (1+t^3) dt, find (f^1)'(0)

1. anonymous

$f(x) = \int\limits_{3}^{x} \sqrt{1+t ^{3}} dt$ find $(f ^{-1})'(0)$

2. anonymous

$f(x) = \int\limits_{3}^{x} \sqrt{1+x ^{3}} dx$ $f(x) = (1+x^3)^{1/2}$ $f'(x) = \frac{ 1 }{ 2 }(1+x ^{3})^{^{-1/2}} (3x ^{2})$

3. slaaibak

f'(x) = sqrt(1+x^3) so f'(0= 1

4. anonymous

is that setting up: putting the 0 from (f^-1)' $0=(1+x^3)^{\frac{ 1 }{ 2 }}$

5. slaaibak

no. so f'(0)= 1 meaning sqrt(1+0^3) = 1

6. anonymous

you just want to sent the equation equal to 0?

7. anonymous

oops set

8. slaaibak

bro, you asked me to find f'(0) and I found f'(0), what are you on about?

9. slaaibak

you know evaluating a function at zero and setting it equal to zero are two different things?

10. anonymous

correct

11. slaaibak

so you gave f(x). I found f'(x) and then I evaluated it at zero (f'(0)) That's the question, isn't it?

12. anonymous

Yes we evaluated f'(x) at zero and we ant to put that into $(f ^{-1})'(0) = \frac{ 1 }{ f'(f ^{-1}(0) }$ Correct? Or am I too tired and my brain is done for the day.

13. anonymous

$= \frac{ 1 }{ f'(1) }$

14. slaaibak

oh! I completely overlooked the inverse over there, sorry

15. anonymous

yeah those lovely inverses

16. anonymous

do we need to back track some for the steps done for doing f'(0) or are we good?

17. slaaibak

so yeah, you solve 0 = f(x) then x=3 do you agree? Then f^-1(0) = 3 ? or alternatively, f(3) = 0

18. anonymous

ok

19. slaaibak

then f'(3) = sqrt(1+3^3) = sqrt(28)

20. anonymous

dang I didn't need to figure that ugly f'(x)?

21. slaaibak

you did. Well, I did. $f'(x) ={d \over dx} \int\limits_{3}^{x} \sqrt{1+t^3}dt = \sqrt{1+x^3}$

22. slaaibak

it's kinda obvious if you think about it. you should get the same answer back, since you are integrating and then differentiating again. BUT there is a chain rule adjustment that has to be made if there are more complex bounds

23. anonymous

I wish these derivative inverses made more sense to me. I am pretty confident in my inverse solving. But these inverse derivatives are killing me.

24. slaaibak

haha, just remember the methods/formulas, do it step by step and you'll be fine

25. anonymous

Thanks for your help slaaibak. Have a great night.

26. slaaibak

No problem. Thanx, you too.

27. anonymous

ok can some one work this problem for me from 1st step to last step. I see you plug x in for t. and 3 in for x but what about the (f^-1)(0) what does that fit in thanks


text:
Sequences

Package of problems

Solution

1. Can you find an arithmetic and a geometric sequence which each have second and fourth terms $u_2=6 \quad\text{and}\quad u_4=54 \text{ ?}$

Let’s start by looking for an arithmetic sequence. We are given the second and fourth terms so the common difference must be $d=\frac{1}{2}(54-6) = 24$ and the first term is $d$ less than the second, so $a=6-24 = -18.$

The arithmetic sequence is $(-18,6,30,54,78,...)$.

The geometric sequence has common ratio, $r$, such that $r^2=\frac{54}{6}=9 \quad\text{and so}\quad r=3.$ Its first term is $6\div3$ and the sequence is $(2,6,18,54,162,...)$.

We can illustrate these two sequences as sets of points on a graph.

We have drawn in lines on which the points lie. Can you describe the shape of the two lines?

Notice that when solving for $r$ above we could also have had $r=-3$. What would our geometric sequence have looked like then?

1. Can you find an arithmetic and a geometric sequence which each have $u_2=-2, \quad u_3=4 \quad\text{and}\quad u_5=16 \text{ ?}$

Having thought about how arithmetic and geometric sequences behave, your first reaction might be that it is impossible to have three terms in common. Let’s have a look at the numbers anyway…

Comparing $u_2$ and $u_3$, the arithmetic sequence must have $d=4-(-2)=6$ and this does fit with $u_5$. The first term, $a=u_2-d=-8$ and the arithmetic sequence is $(-8,-2,4,10,16,22,...)$.

For the geometric sequence, the common ratio must be $r=\frac{u_3}{u_2}=-2$. As a check, $\frac{u_5}{u_3}=4=r^2$ so this does work. The first term will be $\frac{u_2}{r}=1$ and the sequence is $(1,-2,4,-8,16,-32,...)$.

The reason this works is the negative common ratio. This is an example of an oscillating sequence where the values of the terms go above and below a central value, in this case zero.

As before, we can illustrate these sequences with a graph.

The terms of the arithmetic sequence lie along a straight line again, but this time we have had to draw a pair of exponential curves for the geometric sequence – one for the positive terms and one for the negative.

1. A geometric sequence $G$ has first term $8$ and common ratio $-\frac{1}{2}$. Can you find an arithmetic sequence that has the same first term and two other terms the same as the corresponding terms of $G$?

This is similar to the previous example in that $G$ has a negative common ratio and is therefore oscillating. Let’s write out the first few terms. $G=\left(8,-4,2,-1,\frac{1}{2},-\frac{1}{4},...\right)$

We want an arithmetic sequence that has $a=8$ and that meets $G$ again at two other terms. We could try matching some of the early terms and see what happens.

• Try matching the first two terms. Then we have $d=-12$ and the sequence would go $(8,-4,-16,-28,...)$. This is clearly not going to meet $G$ a third time.

• Try the first and third terms. $2d=-6$ so $d=-3$ and the sequence goes $(8,5,2,-1,-4,...)$. This is what we want because its first, third and fourth terms match with $G$.

What does this solution have in common with the solution to (2)? How are they related?

We have found a sequence with the first, third and fourth terms in common with $G$. Do you think that it is possible to find other arithmetic sequences with different sets of three terms in common with $G$?

1. Can you find an arithmetic and a geometric sequence that have four terms in common?

The only way we can get four terms of a geometric sequence to be linearly spaced is if all its terms are identical. A geometric sequence with common ratio $r=1$ and an arithmetic sequence with common difference $d=0$ will have identical terms if their first terms are the same.

You might argue that the sequence $(1,1,1,1,1,...)$ is neither arithmetic nor geometric, but it is generally considered to be both, as it satisfies the definition of both.


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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

IMPACT FACTOR 2018: 1.005

CiteScore 2018: 1.01

SCImago Journal Rank (SJR) 2018: 0.237
Source Normalized Impact per Paper (SNIP) 2018: 0.541

ICV 2018: 147.55

Open Access
Online
ISSN
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Volume 16, Issue 1

# M-polynomials and topological indices of hex-derived networks

Shin Min Kang
• Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 52828, Korea
• Center for General Education, China Medical University, Taichung 40402, Taiwan
• Email
• Other articles by this author:
/ Waqas Nazeer
/ Abdul Rauf Nizami
/ Mobeen Munir
Published Online: 2018-07-17 | DOI: https://doi.org/10.1515/phys-2018-0054

## Abstract

Hex-derived network has a variety of useful applications in pharmacy, electronics, and networking. In this paper, we give general form of the M-polynomial of the hex-derived networksHDN1[n] and HDN2[n], which came out of n-dimensional hexagonal mesh. We also give closed forms of several degree-based topological indices associated to these networks.

PACS: 81.05.-t; 81.07.Nb

## 1 Introduction

In the field of mathematical chemistry, several useful structural and chemical properties of a chemical compound can be determined using simple mathematical tools of polynomials (as Hosoya polynomial and M-polynomial) and numbers (as Randic index and Zagreb index) instead of complicated techniques of quantum mechanics; for details, see Ref. [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].

The hexagonal mesh was introduced by Chen et al. in 1990 [13]. The 2-dimensional hexagonal mesh HX(2), which is composed of six triangles, is given in Figure 1:

Figure 1

2-dimensional hexagonal mesh

If we add a layer of triangles around this 2-dimensional mesh, we obtain the 3-dimensional hexagonal mesh HX(3); see Figure 2:

Figure 2

3-dimensional hexagonal mesh

The n-dimensional hexagonal mesh HX(n) is obtained by attaching n-2 layers of triangles around HX(2).

A connected planar graph G divides the plane into disjoint regions; each region is called the face of G; two faces are said to be adjacent if they have a common edge; the unbounded region lying outside the graph forms the outer face. The 2-dimensional mesh HX(2) has seven faces, as in Figure 3:

Figure 3

Faces of 2-dimansional hexagonal mesh

If corresponding to every closed face f of HX[n] we mark a vertex F and then join it with all the vertices of the face f through edges we receive the hex-derived network HDN1[n]; you can see HDN1 [4] in Figure 4:

Figure 4

The graph of HDN1[4]

If in HDN1[n] the faces f1, f2,…,fk are adjacent to a face f and if the vertex F that represents the face f is joined to the vertices F1, F2, …Fk representing the faces f1, f2, …, fk through edges, we receive the hex-derived network HDN2[n]; one may have a look at HDN2 [4] in Figure 5:

Figure 5

The graph of HDN2[4]

In this report, we give the closed forms of the M-polynomial of HDN1[n] and HDN2[n] and recovered several degree-based topological indices from these polynomials.

Throughout this paper, G will represent a connected graph, V its vertex set, E its edge set, and dv the degree of its vertex v.

#### Definition 1

The M-polynomial of G is defined as

$MG,x,y=∑δ≤i≤j≤ΔmijGxiyj,$(1)

where δ = Min{dv|v ∈ V(G)}, Δ = Max{dv|v ∈ V(G)}, and mij(G) is the edge vuE(G) such that {dv, du} = {i, j}.

Topological indices are graph invariants and presently play an important role in the field of mathematical chemistry, biology, physics, electronics and other applied areas. So for many useful topological indices have been introduced. In 1975, Milan Randic´ introduced the Randic´ index [18], which is defined as

$R−1/2(G)=∑uv∈E(G)1dudv.$(2)

For a reasonable information about the development and applications of the Randić index, we refer to [24, 25, 26, 27, 28, 29, 30, 31].

In 1998, working independently, Bollobas and Erdos [19] and Amic et al. [20] proposed the generalized Randic´ index; see [21, 22, 23] for more information.

The general Randic´ index is defined as

$Rα(G)=∑uv∈E(G)(dudv)α,$(3)

and the inverse Randic´ index is defined as RRα(G) = $\begin{array}{}\sum _{uv\in E\left(G\right)}\frac{1}{{\left({d}_{u}{d}_{v}\right)}^{\alpha }}.\end{array}$

Gutman and Trinajstic´ introduced first Zagreb index and second Zagreb index, which are respectively M1(G) = $\begin{array}{}\sum _{uv\in E\left(G\right)}\end{array}$ (du + dv) and M2(G) = $\begin{array}{}\sum _{uv\in E\left(G\right)}\end{array}$ (du × dv).

The second modified Zagreb index is defined as

$mM2G=∑uv∈E(G)1d(u)d(v).$(4)

For detailed information, we refer the reader to [32, 33, 34, 35, 36].

The symmetric division index is

$SDDG=∑uv∈E(G)min(du,dv)max(du,dv)+max(du,dv)min(du,dv).$(5)

Another variant of Randic index is the harmonic index, which is defined as

$H(G)=∑vu∈E(G)2du+dv.$(6)

The inverse-sum index is

$I(G)=∑vu∈E(G)dudvdu+dv.$(7)

The augmented Zagreb index is

$A(G)=∑vu∈E(G)dudvdu+dv−23,$(8)

which is found useful for computing the heat of formation of alkanes [37, 38].

Some well-known degree-based topological indices are closely related to the M-polynomial [7]; in Table 1 you can see such relations.

Table 1

Derivation of some degree-based topological indices from M-polynomial

Here

$Dx=x∂(fx,y∂x,Dy=y∂(fx,y∂y,Sx=∫0xft,ytdt,Sy=∫0yfx,ttdtJfx,y=fx,x,Qαfx,y=xαfx,y.$(9)

## 2 Main results

This section contains the general closed forms of the M polynomial and related indices of the hex-derived networks HDN1[n] and HDN2[n].

#### Theorem 1

The M-polynomial of HDN1[n], n > 3, is

$MHDN1[n];x,y=12x3y5+(18n−36)x3y7+(18n2−54n+42)x3y12+12nx5y7+6x5y12+(6n−18)x7y7+(12n−24)x7y12+(9n2−33n+30)x12y12.$(10)

#### Proof

Depending on degrees, the vertex set E of HDN1[n] can be divided into four disjoint subsets, V1, V2, V3, and V4, containing vertices of degrees 3, 5, 7, and 12, respectively.

Moreover, |V1(HDN1[n])| = 6n2–12n+6, |V2(HDN1[n])| = 6, |V3(HDN1[n])| = 6n – 12, and |V4(HDN1[n])| = 3n2 – 9n + 7.

Similarly, the edge set E of HDN1[n] can be divided into eight disjoint subsets:

$E1(HDN1[n])=e=uv∈E(HDN1[n]):du=3,dv=5,E2(HDN1[n])=e=uv∈E(HDN1[n]):du=3,dv=7,E3(HDN1[n])=e=uv∈E(HDN1[n]):du=3,dv=12,E4(HDN1[n])=e=uv∈E(HDN1[n]):du=5,dv=7,E5(HDN1[n])=e=uv∈E(HDN1[n]):du=5,dv=12,E6(HDN1[n])=e=uv∈E(HDN1[n]):du=7,dv=7,E7(HDN1[n])=e=uv∈E(HDN1[n]):du=7,dv=12,E8(HDN1[n])=e=uv∈E(HDN1[n]):du=dv=12.$

Also,

$E1(HDN1[n])=12, E2(HDN1[n])=18n−36,E3(HDN1[n])=18n2−54n+42,E4(HDN1[n])=12,E5(HDN1[n])=6,$

$E6(HDN1[n])=6n−18,$(11)

$E7(HDN1[n])=12n−24,$(12)

$E8(HDN1[n])=9n2−33n+30.$(13)

Now, we have

$MHDN1[n];x,y=∑i≤jmi,jxiyj=∑3≤5m3,5x3y5+∑3≤7m3,7x3y7+∑3≤12m3,12x3y12+∑5≤7m5,7x5y7+∑5≤12m5,12x5y12+∑7≤7m7,7x7y7+∑7≤12m7,12x7y12+∑12≤12m12,12x12y12=∑uv∈E1(HDN1[n])m3,5x3y5+∑uv∈E2(HDN1[n])m3,7x3y7+∑uv∈E3(HDN1[n])m3,12x3y12+∑uv∈E4(HDN1[n])m5,7x5y7+∑uv∈E5(HDN1[n])m5,12x5y12+∑uv∈E6(HDN1[n])m7,7x7y7+∑uv∈E7(HDN1[n])m7,12x7y12+∑uv∈E8(HDN1[n])∑12≤12m12,12x12y12=E1(HDN1[n])x3y5+E2(HDN1[n])x3y7+E3(HDN3[n])x3y12+E4(HDN1[n])x5y7+E5(HDN1[n])x5y12+E6(HDN1[n])x7y7+E7(HDN1[n])x7y12+E8(HDN1[n])x12y12=12x3y5+(18n−36)x3y7+(18n2−54n+42)x3y12+12nx5y7+6x5y12+(6n−18)x7y7+(12n−24)x7y12+(9n2−33n+30)x12y12.$(35)

Some degree-based topologcal indices of HDN1[n] are given in the following proposition.

#### Proposition 1

For the hex-derived network HDN1[n], we have

1. M1(HDN1[n]) = 486n2 – 966n + 480.

2. M2(HDN1[n]) = 1944n2 – 4596n + 2718.

3. $\begin{array}{}{}^{m}{M}_{2}\left(HD{N}_{1}\left[n\right]\right)=-\frac{181}{1960}-\frac{3103}{11760}n+\frac{9}{16}{n}^{2}.\end{array}$

4. $\begin{array}{}Kk\begin{array}{c}R{R}_{\alpha }\left(HD{N}_{1}\left[n\right]\right)=12×{15}^{\alpha }+\left(18n-36\right){21}^{\alpha }\\ +\left(18{n}^{2}-54n+42\right){36}^{\alpha }+12n{35}^{\alpha }+6×{60}^{\alpha }\\ +\left(6n-18\right){49}^{\alpha }+\left(12n-24\right){84}^{\alpha }\\ +\left(9{n}^{2}-33n+30\right){144}^{\alpha }.\end{array}kk\end{array}$

5. $\begin{array}{}{R}_{\alpha }\left(HD{N}_{1}\left[n\right]\right)=\frac{12}{{15}^{\alpha }}+\frac{18n-36}{{21}^{\alpha }}+\frac{18{n}^{2}-54n+42}{{36}^{\alpha }}\\ +\frac{12n}{{35}^{\alpha }}+\frac{6}{{60}^{\alpha }}+\frac{6n-18}{{49}^{\alpha }}+\frac{12n-24}{{84}^{\alpha }}+\frac{9{n}^{2}-33n+30}{{144}^{\alpha }}.\end{array}$

6. $\begin{array}{}SSD\left(HD{N}_{1}\left[n\right]\right)=\frac{3221}{35}-\frac{12659}{70}n+\frac{189}{2}{n}^{2}.\end{array}$

7. $\begin{array}{}H\left(HD{N}_{1}\left[n\right]\right)=-\frac{11121}{45220}-\frac{5931}{5320}n+\frac{63}{40}{n}^{2}.\end{array}$

8. $\begin{array}{}I\left(HD{N}_{1}\left[n\right]\right)=\frac{257661}{3230}-\frac{17171}{95}n+\frac{486}{5}{n}^{2}.\end{array}$

9. $\begin{array}{}A\left(HD{N}_{1}\left[n\right]\right)=\frac{23249577512627267}{5516785532544}\\ -\frac{19741483367980655}{3009155745024}n+\frac{7382820168}{2924207}{n}^{2}.\end{array}$

#### Proof

Consider the M-polynomial of HDN1[1], which is

$M(HDN1[n];x,y)=f(x,y)=12x3y5+(18n−36)x3y7+(18n2−54n+42)x3y12+12nx5y7+6x5y12+(6n−18)x7y7+(12n−24)x7y12+(9n2−33n+30)x12y12.$

Then

$Dxf(x,y)=36x3y5+3(18n−36)x3y7+3(18n2−54n+42)x3y12+60nx5y7+30x5y12+7(6n−18)x7y7+7(12n−24)x7y12+12(9n2−33n+30)x12y12,$

$Dyf(x,y)=60x3y5+7(18n−36)x3y7+12(18n2−54n+42)x3y12+84nx5y7+72x5y12+7(6n−18)x7y7+12(12n−24)x7y12+12(9n2−33n+30)x12y12,$

$DyDxf(x,y)=180x3y5+21(18n−36)x3y7+36(18n2−54n+42)x3y12+420nx5y7+360x5y12+49(6n−18)x7y7+84(12n−24)x7y12+144(9n2−33n+30)x12y12,$

$Sy(f(x,y))=125x3y5+17(18n−36)x3y7+112(18n2−54n+42)x3y12+127nx5y7+12x5y12+17(6n−18)x7y7+112(12n−24)x7y12+112(9n2−33n+30)x12y12,$

$SxSy(f(x,y))=1215x3y5+121(18n−36)x3y7+136(18n2−54n+42)x3y12+1235nx5y7+110x5y12+149(6n−18)x7y7+184(12n−24)x7y12+1144(9n2−33n+30)x12y12,$

$DxαDyα(f(x,y))=12×3α5αx3y5+3α7α(18n−36)x3y7+3α12α(18n2−54n+42)x3y12+12×5α7αnx5y7+6×5α12αx5y12+72α(6n−18)x7y7+7α12α(12n−24)x7y12+122α(9n2−33n+30)x12y12,$

$SyDxf(x,y)=365x3y5+37(18n−36)x3y7+14(18n2−54n+42)x3y12+607nx5y7+3012x5y12+(6n−18)x7y7+712(12n−24)x7y12+(9n2−33n+30)x12y12,$

$SxDyf(x,y)=20x3y5+73(18n−36)x3y7+4(18n2−54n+42)x3y12+845nx5y7+725x5y12+(6n−18)x7y7+127(12n−24)x7y12+(9n2−33n+30)x12y12,$

$SxJf(x,y)=32x8+110(18n−36)x10+115(18n2−54n+42)x15+nx12+617x17+114(6n−18)x14+119(12n−24)x19+124(9n2−33n+30)x24,$

$SxJDxDyf(x,y)=452x8+2110(18n−36)x10+3615(18n2−54n+42)x15+35nx12+36017x17+4914(6n−18)x14+8419(12n−24)x19+6(9n2−33n+30)x24,$

$Sx3Q−2JDx3Dy3f(x,y)=12×335363x6+337383(18n−36)x8+33123133(18n2−54n+42)x13+12×5373103nx10+6×53123153x15+76(6n−18)123x12+73123173(12n−24)x17+126223(9n2−33n+30)x22,$

Now, we go for indices.

1. M1(HDN1[n]) = Dx + Dy(f(x, y))|x=y=1 = 486n2 − 966n + 480

2. M2(HDN1[n]) = DxDy(f(x, y))|x=y=1 = 1944n2 − 4596n + 2718.

3. $\begin{array}{}{}^{m}{M}_{2}\left(HD{N}_{1}\left[n\right]\right)={{S}_{x}{S}_{y}\left(f\left(x,y\right)\right)|}_{x=y=1}=-\frac{181}{1960}& \\ -\frac{3103}{11760}n+\frac{9}{16}{n}^{2}.\end{array}$

4. $\begin{array}{}{R}_{\alpha }\left(HD{N}_{1}\left[n\right]\right)={{D}_{x}^{\alpha }{D}_{y}^{\alpha }\left(f\left(x,y\right)\right)|}_{x=y=1}=12×{15}^{\alpha }\\ +\left(18n-36\right){21}^{\alpha }+\left(18{n}^{2}-54n+42\right){36}^{\alpha }+12n{35}^{\alpha }\\ +6×{60}^{\alpha }+\left(6n-18\right){49}^{\alpha }+\left(12n-24\right){84}^{\alpha }\\ +\left(9{n}^{2}-33n+30\right){144}^{\alpha }.\end{array}$

5. $\begin{array}{}R{R}_{\alpha }\left(HD{N}_{1}\left[n\right]\right)={{S}_{x}^{\alpha }{S}_{y}^{\alpha }\left(f\left(x,y\right)\right)|}_{x=y=1}=\frac{12}{{15}^{\alpha }}& \\ +\frac{18n-36}{{21}^{\alpha }}+\frac{18{n}^{2}-54n+42}{{36}^{\alpha }}+\frac{12n}{{35}^{\alpha }}+\frac{6}{{60}^{\alpha }}\\ +\frac{6n-18}{{49}^{\alpha }}+\frac{12n-24}{{84}^{\alpha }}+\frac{9{n}^{2}-33n+30}{{144}^{\alpha }}.\end{array}$

6. $\begin{array}{}SSD\left(HD{N}_{1}\left[n\right]\right)={\left({S}_{y}{D}_{x}+{S}_{x}{D}_{y}\right)\left(f\left(x,y\right)\right)|}_{x=y=1}\\ =\frac{3221}{35}-\frac{12659}{70}n+\frac{189}{2}{n}^{2}.\end{array}$

7. $\begin{array}{}H\left(HD{N}_{1}\left[n\right]\right)=2{S}_{x}J\left(f\left(x,y\right)\right){|}_{x=1}=-\frac{11121}{45220}\\ -\frac{5931}{5320}n+\frac{63}{40}{n}^{2}.\end{array}$

8. $\begin{array}{}I\left(HD{N}_{1}\left[n\right]\right)={S}_{x}J{D}_{x}{D}_{y}{\left(f\left(x,y\right)\right)}_{x=1}=\frac{257661}{3230}\\ -\frac{17171}{95}n+\frac{486}{5}{n}^{2}.\end{array}$

9. $\begin{array}{}A\left(HD{N}_{1}\left[n\right]\right)={{{S}_{x}}^{3}{Q}_{-2}J{{D}_{x}}^{3}{{D}_{y}}^{3}\left(f\left(x,y\right)\right)|}_{x=1}\\ =\frac{23249577512627267}{5516785532544}-\frac{19741483367980655}{3009155745024}n\\ +\frac{7382820168}{2924207}{n}^{2}.\end{array}$

#### Theorem 2

The M-polynomial of HDN2[n] is

$MHDN2[n];x,y=18x5y5+(12n−24)x5y6+(12n−12)x5y7+6nx5y12+(9n2−33n+30)x6y6+(6n−12)x6y7+(18n2−60n+48)x6y12+(6n−18)x7y7+(12n−24)x7y12+(9n2−33n+30)x12y12.$

#### Proof

Consider the hexagonal mesh HDN2[n], where n > 3. The vertex set of HDN2[n] has the following four partitions.

$V1(HDN2[n])=u∈V(HDN2[n]):du=5,$(48)

$V2(HDN2[n])=u∈V(HDN2[n]):du=6,$(49)

$V3(HDN2[n])=u∈V(HDN2[n]):du=7,$(50)

and

$V4(HDN2[n])=u∈V(HDN2[n]):du=12.$(51)

Also

$V1(HDN2[n])=6n,$(52)

$V2(HDN2[n])=6n2−−18n+12,$(53)

$V3(HDN2[n])=6n−−12,$(54)

$V4(HDN2[n])=3n2−−9n+7.$(55)

Moreover we divide the edge set of HDN2[n] into the following ten partitions:

$E1(HDN2[n])=e=uv∈E(HDN2[n]):du=dv=5,$(56)

$E2(HDN2[n])=e=uv∈E(HDN2[n]):du=5,dv=6,$(57)

$E3(HDN2[n])=e=uv∈E(HDN2[n]):du=5,dv=7,$(58)

$E4(HDN2[n])=e=uv∈E(HDN2[n]):du=5,dv=12,$(59)

$E5(HDN2[n])=e=uv∈E(HDN2[n]):du=6,dv=6,$(60)

$E6(HDN2[n])=e=uv∈E(HDN2[n]):du=6,dv=7,$(61)

$E7(HDN2[n])=e=uv∈E(HDN2[n]):du=6,dv=12,$(62)

$E8(HDN2[n])=e=uv∈E(HDN2[n]):du=7,dv=7,$(63)

$E9(HDN2[n])=e=uv∈E(HDN2[n]):du=7,dv=12,$(64)

and

$E10(HDN2[n])=e=uv∈E(HDN2[n]):du=12,dv=12,$(65)

Also, we have

$E1(HDN2[n])=18$(66)

$E2(HDN2[n])=12n−24,$(67)

$E3(HDN2[n])=12n−12,$(68)

$E4(HDN2[n])=6n,$(69)

$E5(HDN2[n])=9n2−33n+30,$(70)

$E6(HDN2[n])=6n−12,$(71)

$E7(HDN2[n])=18n2−60n+48,$(72)

$E8(HDN2[n])=6n−18,$(73)

$E9(HDN2[n])=12n−24,$(74)

and

$E10(HDN2[n])=9n2−33n+30.$(75)

Now, by the definition of the M-polynomial, we have

$M(HDN2[n];x,y)=∑i≤jmi,jxiyj=∑5≤5m5,5x5y5+∑5≤6m5,6x5y6+∑5≤7m5,7x5y7+∑5≤12m5,12x5y12+∑6≤6m6,6x6y6+∑6≤7m6,7x6y7+∑6≤12m6,12x6y12+∑7≤7m7,7x7y7+∑7≤12m7,12x7y12+∑12≤12m12,12x12y12=∑uv∈E1(HDN2[n])m5,5x5y5+∑uv∈E2(HDN2[n])m5,6x5y6+∑uv∈E3(HDN2[n])m5,7x5y7+∑uv∈E4(HDN2[n])m5,12x5y12+∑uv∈E5(HDN2[n])m6,6x6y6+∑uv∈E6(HDN2[n])m6,7x6y7+∑uv∈E7(HDN2[n])m6,12x6y12+∑uv∈E8(HDN2[n])m7,7x7y7+∑uv∈E9(HDN2[n])m7,12x7y12+∑uv∈E10(HDN2[n])m12,12x12y12=E1(HDN2[n])x5y5+E2(HDN2[n])x5y6+E3(HDN2[n])x5y7+E4(HDN2[n])x5y12+E5(HDN2[n])x6y6+E6(HDN2[n])x6y7+E7(HDN2[n])x6y12+E8(HDN2[n])x7y7+E9(HDN2[n])x7y12+E10(HDN2[n])x12y12=18x5y5+(12n−24)x5y6+(12n−12)x5y7+6nx5y12+(9n2−33n+30)x6y6+(6n−12)x6y7+(18n2−60n+48)x6y12+(6n−18)x7y7+(12n−24)x7y12+(9n2−33n+30)x12y12.$(76)

Now we compute some degree-based topological indices of the hexagonal mesh.

#### Proposition 2

Forwe have

1. M1(HDN2[n]) = 648n2 − 1500n + 852.

2. M2(HDN2[n]) = 2916n2 − 7566n + 4764.

3. $\begin{array}{}{}^{m}{M}_{2}\left(HD{N}_{2}\left[n\right]\right)=\frac{10193}{29400}-\frac{8586}{11760}n+\frac{9}{16}{n}^{2}.\end{array}$

4. $\begin{array}{}{R}_{\alpha }\left(HD{N}_{2}\left[n\right]\right)=18×{25}^{\alpha }+\left(12n-24\right){30}^{\alpha }\\ +\left(12n-12\right){35}^{\alpha }+6n{60}^{\alpha }+\left(9{n}^{2}-33n+30\right){36}^{\alpha }\\ +\left(6n-12\right){42}^{\alpha }+\left(18{n}^{2}-60n+48\right){72}^{\alpha }\\ +\left(6n-18\right){49}^{\alpha }+\left(12n-24\right){84}^{\alpha }\\ +\left(9{n}^{2}-33n+30\right){144}^{\alpha }.\end{array}$

5. $\begin{array}{}R{R}_{\alpha }\left(HD{N}_{2}\left[n\right]\right)=\frac{18}{{25}^{\alpha }}+\frac{12n-24}{{30}^{\alpha }}+\frac{12n-12}{{35}^{\alpha }}\\ +\frac{6n}{{60}^{\alpha }}+\frac{9{n}^{2}-33n+30}{{36}^{\alpha }}+\frac{6n-12}{{42}^{\alpha }}\\ +\frac{18{n}^{2}-60n+48}{{72}^{\alpha }}+\frac{6n-18}{{49}^{\alpha }}+\frac{12n-24}{{84}^{\alpha }}\\ +\frac{9{n}^{2}-33n+30}{{144}^{\alpha }}.\end{array}$

6. $\begin{array}{}SSD\left(HD{N}_{2}\left[n\right]\right)=-\frac{11453}{70}n+\frac{432}{5}+81{n}^{2}.\end{array}$

7. $\begin{array}{}H\left(HD{N}_{2}\left[n\right]\right)=\frac{1783487}{1141140}-\frac{27103217}{7759752}n+\frac{17}{8}{n}^{2}.\end{array}$

8. $\begin{array}{}I\left(HD{N}_{2}\left[n\right]\right)=\frac{539789}{2717}-\frac{16381345}{46189}n+153{n}^{2}.\end{array}$

9. $\begin{array}{}A\left(HD{N}_{2}\left[n\right]\right)=\frac{3218008599887179}{376658092800}\\ -\frac{998781364704403}{78470436000}n+\frac{3050648487}{665500}{n}^{2}.\end{array}$

#### Proof

Let

$fx,y=18x5y5+(12n−24)x5y6+(12n−12)x5y7+6nx5y12+(9n2−33n+30)x6y6+(6n−12)x6y7+(18n2−60n+48)x6y12+(6n−18)x7y7+(12n−24)x7y12+(9n2−33n+30)x12y12.$(77)

Then

$Dyf(x,y)=90x5y5+6(12n−24)x5y6+7(12n−12)x5y7+72nx5y12+6(9n2−33n+30)x6y6+7(6n−12)x6y7+12(18n2−60n+48)x6y12+7(6n−18)x7y7+12(12n−24)x7y12+12(9n2−33n+30)x12y12,$(78)

$Dyf(x,y)=90x5y5+6(12n−24)x5y6+7(12n−12)x5y7+72nx5y12+6(9n2−33n+30)x6y6+7(6n−12)x6y7+12(18n2−60n+48)x6y12+7(6n−18)x7y7+12(12n−24)x7y12+12(9n2−33n+30)x12y12,$(79)

$DyDx(f(x,y))=450x5y5+(12n−24)x5y6+35(12n−12)x5y7+360nx5y12+36(9n2−33n+30)x6y6+42(6n−12)x6y7+72(18n2−60n+48)x6y12+49(6n−18)x7y7+84(12n−24)x7y12+144(9n2−33n+30)x12y12,$(80)

$SxSy(f(x,y))=1825x5y5+130(12n−24)x5y6+135(12n−12)x5y7+110nx5y12+136(9n2−33n+30)x6y6+142(6n−12)x6y7+172(18n2−60n+48)x6y12+149(6n−18)x7y7+184(12n−24)x7y12+1144(9n2−33n+30)x12y12,$(81)

$DxαDyα(f(x,y))=18×52αx5y5+5α6α(12n−24)x5y6+5α7α(12n−12)x5y7+6×5α12αnx5y12+62α(9n2−33n+30)x6y6+6α7α(6n−12)x6y7+6α12α(18n2−60n+48)x6y12+72α(6n−18)x7y7+7α12α(12n−24)x7y12+122α(9n2−33n+30)x12y12,$(82)

$SxαSyα(f(x,y))=1852αx5y5+15α6α(12n−24)x5y6+15α7α(12n−12)x5y7+65α12αnx5y12+162α(9n2−33n+30)x6y6+16α7α(6n−12)x6y7+16α12α(18n2−60n+48)x6y12+172α(6n−18)x7y7+17α12α(12n−24)x7y12+1122α(9n2−33n+30)x12y12,$(83)

$SyDx(f(x,y))=18x5y5+56(12n−24)x5y6+57(12n−12)x5y7+52nx5y12+(9n2−33n+30)x6y6+67(6n−12)x6y7+12(18n2−60n+48)x6y12+(6n−18)x7y7+712(12n−24)x7y12+(9n2−33n+30)x12y12,$(84)

$SxDy(f(x,y))=18x5y5+6(12n−24)x5y6+75(12n−12)x5y7+725nx5y12+(9n2−33n+30)x6y6+76(6n−12)x6y7+2(18n2−60n+48)x6y12+(6n−18)x7y7+127(12n−24)x7y12+(9n2−33n+30)x12y12,$(85)

$SxJf(x,y)=95x10+111(12n−24)x11+(n−1)x12+617nx17+112(9n2−33n+30)x12+113(6n−12)x13+118(18n2−60n+48)x18+114(6n−18)x14+119(12n−24)x19+124(9n2−33n+30)x24,$(86)

$SxJDxDyf(x,y)=45x10+3011(12n−24)x11+3512(12n−12)x12+36017nx17+3(9n2−33n+30)x124213(6n−12)x13+4(18n2−60n+48)x18+4914(6n−18)x14+8419(12n−24)x19+6(9n2−33n+30)x24,$(87)

$Sx3Q−2JDx3Dy3f(x,y)=94×56x8+536393(12n−24)x9+5373103(12n−12)x10+6×53123153nx15+66103(9n2−33n+30)x10+6373113(6n−12)x11+63123163(18n2−60n+48)x16+76123(6n−18)x12+73123173(12n−24)x17+126223(9n2−33n+30)x22.$(88)

1. $M1(HDN2[n])=Dx+Dy(f(x,y))|x=y=1=648n2−1500n+852.$

2. $M2(HDN2[n])=DxDy(f(x,y))|x=y=1=2916n2−7566n+4764.$

3. $mM2(HDN2[n])=SxSy(f(x,y))|x=y=1=1019329400−856311760n+916n2.$

4. $Rα(HDN2[n])=DxαDyα(f(x,y))|x=y=1=18×25α+(12n−24)30α+(12n−12)35α+6n60α+(9n2−33n+30)36α+(6n−12)42α+(18n2−60n+48)72α+(6n−18)49α+(12n−24)84α+(9n2−33n+30)144α.$

5. $RRα(HDN2[n])=SxαSyα(f(x,y))|x=y=1=1825α+12n−2430α+12n−1235α+6n60α+9n2−33n+3036α+6n−1242α+18n2−60n+4872α+6n−1849α+12n−2484α+9n2−33n+30144α.$

6. $SSD(HDN2[n])=(SyDx+SxDy)(f(x,y))|x=y=1=−1145370n+4325+81n2.$

7. $H(HDN2[n])=2SxJ(f(x,y))|x=1=17834871141140−271032177759752n+178n2.$

8. $I(HDN2[n])=SxJDxDy(f(x,y))x=1=5397892717−1638134546189n+153n2.$

9. $A(HDN2[n])=Sx3Q−2JDx3Dy3(f(x,y))|x=1=3218008599887179376658092800−99878136470440378470436000n+3050648487665500n2.$

## 3 Conclusion

In this article, we computed the M-polynomial of and HDN2(n). The First and the second Zagreb indices, Generalized Randic index, Inverse Randic index, Symmetric division index, Inverse sum index and Augmented Zagreb index of these hex-derived networks have also been computed. These indices are actually functions of chemical graphs and encode many chemical properties as viscosity, strain energy, and heat of formation. Graphical description, given in Figures 6 and 7, also demonstrates the behavior of the M-polynomial of the networks. It is notable that our results about Randic index extend the results given in [39].

Figure 6

The plot for the M-polynomial of HDN1[1]

Figure 7

The plot for the M-polynomial of HDN2[1]

## Acknowledgement

This research is supported by Higher Education Commission of Pakistan.

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Tel.: +923214707379

Accepted: 2018-04-27

Published Online: 2018-07-17

Autor Contributions: All authors contributed equally in writing this paper.

Conflict of InterestConflict of Interests: The authors declare no conflict of interest.

Citation Information: Open Physics, Volume 16, Issue 1, Pages 394–403, ISSN (Online) 2391-5471,

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text:
# Why RMSE is too small?

Sorry, I'm a newbie at recommender systems, but I wrote few lines of code using the Apache Mahout library.

My dataset is pretty small, 500x100 with 8102 cells known.

The dataset is actually a subset of the Yelp dataset from the "Yelp business rating prediction" competition. I just took the top 100 most commented restaurants, and then took the 500 most active customers.

I created SVDRecommender and then I evaluated RMSE.

The result is about 0.4... Why is it so small?

Maybe I just don't understand something and my dataset is not so sparse, but then I tried with a larger and more sparse dataset and RMSE became even smaller (about 0.18)! Could anyone explain this behaviour?

DataModel model = new FileDataModel(new File("datamf.csv"));
final RatingSGDFactorizer factorizer = new RatingSGDFactorizer(model, 20, 200);
final Factorization f = factorizer.factorize();

RecommenderBuilder builder = new RecommenderBuilder() {
public Recommender buildRecommender(DataModel model) throws TasteException {
return new SVDRecommender(model, factorizer);
}
};

RecommenderEvaluator evaluator = new RMSRecommenderEvaluator();
double score = evaluator.evaluate(builder,
null,
model,
0.6,
1);

System.out.println(score);

• RMSE is relative measure, i.e. it is expressed in the scale of your variable (see e.g. here). So there is no "objectively small" value for RMSE, it depends on your data. So you have to tell us why do you consider this value to be "too small"? What is your data? – Tim Jun 20 '15 at 9:04
• @Tim. Well, yeah, thanks, i thought that RMSE is an absolute measure. But how can I interpret this result? Should I calculate the correlation coefficient? My data, as I said, is a subset of Yelp dataset 500x100 with 8102 cells defined. Each cell is a rating from 1 to 5. – luckyfish Jun 20 '15 at 9:18
• You use measures such as RMSE to compare and/or adjust models for the same data, so if you have only one model and one RMSE value does not really tell you much. – Tim Jun 20 '15 at 9:26
• @Tim As I understand, RMSE strongly depends on data? I thought that RMSE is an absolute value, so i can say that "my recommender predict ratings with rmse_value error". and now, as i understand, I should talk "my recommender predicts ratings with this error on this data"? What are the data characteristics RMSE depends on (size, sparsity or scale of rating values)? (sorry for my bad english) – luckyfish Jun 20 '15 at 9:50
• Check the link I posted in in the first comment - in describes how RMSE is computed and so tells you on what it depends exactly. I would also recommend you come book on machine learning, see stats.stackexchange.com/questions/143402/… or stats.stackexchange.com/questions/18973/… for references. – Tim Jun 20 '15 at 10:04