Representations of graphs
A graph is a mathematical structure that is sometimes hard to separate from its visualization. An important branch of graph theory studies graph drawing problems. Recently a mathematical approach to graph visualization has been developed under the name of "graph representations". In this tutorial we present an outline of the theory of graph representations. A graph representation is a mapping from the set of vertices of a graph into some representation space. In general, a representation may be degenerate in various well-defined senses. Sometimes, only non-degenerate representations, or realizations are sought. If the representation space has the structure of a metric space, it is possible to define the energy of a representation. This gives rise to a number of optimization problems about graph representations which have been known in the past under names such as molecular mechanics, spring embedding algorithms, etc. Representations of graphs may exhibit some of the graph symmetry. Representations of graphs exhibiting specific graph symmetry are usually quite interesting to find. Finally, graphs themselves are frequently used for representing other mathematical structures, such as networks, posets, polytopes, maps, tillings, configurations, etc. This, in turn, opens up a possibility of a theory of representations for these structures.