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Paper text:
averaging over different graphs is basic in numerous applications of the graph theory @xcite . for such tasks , knowledge of
the number of topologically different graphs is of primary importance . having two graphs ,
a typical question is : are they different ? if the graphs are labeled , respective algorithms are of polynomial time . however , for unlabeled graphs the task should be to check all possible labellings , what makes the problem unfeasible @xcite .
an alternative solution is to find a quantity which is different for different graphs , and of the same value if the graphs are topologically equivalent .
the latter means that there is a one - to - one transformation from one graph to another : each pair of nodes linked ( not linked ) in one graph is linked ( not linked ) in another graph .
such a quantity is a topological invariant .
however , actually we can be never sure if the quantity proposed as the invariant has indeed the above discriminating property .
while its different values certainly mean different graph structures , the same value does not allow to claim that the graphs are indeed topologically identical .
in many cases , the proposed quantity appears to be degenerate , i.e. its value is the same for different graphs .
all that remains true for unlabeled trees , which are graphs without cyclic paths and without loops .    in a series of papers , schultz et al . proposed and evaluated some scalar quantities as candidates to be topological invariants for trees @xcite .
this work was motivated by a chemical application of the constructed quantities , which were found to increase monotonically with the melting temperature of alkanes .
however , almost all proposed invariants were found to be degenerate . on the other hand ,
the last proposed invariant is a real number and not integer , and the comparison of its value must rely on the numerical accuracy .
here we propose a new candidate as a topological invariant for unlabeled trees .
unlike the quantities discussed previously , this is a set of matrices and not a single number .
the advantage is that the matrices are ordered in a simple way , and the ordering algorithm works in polynomial time . on the other hand , to state that two trees are topologically identical we compare all the matrix elements .
this modification is expected to enhance the discriminative force of the proposed invariant .
we use the obtained criterion to calculate the number of topologically non - equivalent trees up to @xmath7 nodes .
as stated above , the obtained numbers can be treated only as an evaluation of the true results from below .
then , if one has a better criterion , he should find the greater number of trees for @xmath8 , than our result , given in table 1 .
.[tab ] the number of trees @xmath9 evaluated basing on sorted @xmath10 pairs with @xmath11 .
@xmath12 is given by the otter s formula .
[ cols= " < , > , > , > , > , > , > , > , > , > , > " , ]     the purpose of introducing mti was to differentiate chemical molecules . when a carbon atom ( with proper number of hydrogen atoms ) is assigned to all nodes of trees shown in fig .
[ fig - missing](c)-(f ) they may represent semi - structural formulas of ( c ) 2,2,4-trimethylpentane , ( d ) 3-ethyl-2-methylpentane , ( e ) 2,2-dimethylhexane and ( f ) 3-ethylhexane @xcite . the mti can not differ between pairs ( c , d ) and ( e , f ) of these forms of @xmath13 .
our results contain not only the number of trees , but the structure of all of them .
binary files with distance matrices and the program for their conversion to input files for pajek @xcite program are available from our web page @xcite .
now we are going to prove that for large @xmath0 , the range of any discriminative topological invariant with integer values should increase exponentially with @xmath0 . to each tree
, a different value of the invariant must be assigned , if the invariant is discriminative
. then we get an exponentially increasing number of different integer values .
the length of a range on an axis , where these values can be placed , must increase also at least exponentially , what finishes the proof .
we note that the matrix character of the invariant does not change this result , as long as the matrix size increases as @xmath14 , where @xmath15 is a constant . in our case @xmath16 , because the matrix is @xmath1 .
we should add that this ` range criterion ' is crucial in the asymptotic regime of large @xmath0 .
up to now , the computational resources do not allow to penetrate this region .
concluding , we have proposed a new topological invariant to discriminate unlabeled trees .
the matrix character of the invariant allows to believe , that the discriminating power of the invariant is much better , than scalar invariants proposed previously .
thanks aleksandra jung for her valuable help .
calculations were carried out in ack - cyfronet - agh .
the machine time on sgi 2800 is financed by the polish ministry of science and information technology under grant no .
kbn / sgi2800/agh/018/2003 .
we propose a new topological invariant of unlabeled trees of @xmath0 nodes . 
 the invariant is a set of @xmath1 matrices of integers , with @xmath2 and @xmath3 as the matrix elements , where @xmath4 are the elements of the distance matrix and @xmath3 denotes @xmath5-th node s degree and @xmath6 . to compare the invariant calculated for possibly different graphs , 
 the matrix rows are ordered with respect to first column , and  if necessary  with respect to the second one . 
 we use the new invariant to evaluate from below the number of topologically different unlabeled trees up to @xmath7 . 
 the results slightly exceed the asymptotic evaluation of otter 
 .    computer modeling and simulation , trees and graphs