On the application of some graph invariants in communication network research tasks

Abstract

The problem of checking the possible isomorphism of graphs has a wide practical application and is an important problem for theoretical computer science in general and the theory of algorithms in particular. Among the numerous areas of application of algorithms for solving the problem of determining graph isomorphism, we note the problem of syntactic and structural pattern recognition, some problems of mathematical chemistry and chemoinformatics (study of molecular structures of chemical compounds), problems related to the study of social networks (for example, linking several accounts of one user on Facebook). It is not known whether it belongs to a polynomial algorithm for this problem — assuming P≠NP. It is known, for example, that NP-complete is the related problem of finding an isomorphic subgraph in a given graph. Thus, the current research is relevant, which is aimed at solving the problem of checking isomorphism for both arbitrary graphs and graphs of a special type (in practice, both exact and heuristic algorithms can be used for such studies). In various algorithms for working with graphs, one of the most common invariants is the vector of degrees. However, the use of this invariant alone for constructing most practical algorithms on graphs is apparently not sufficient; its possible generalization is the more complex invariant considered by the authors — the vector of second–order degrees. At the same time, the graphs considered in this paper with the generated vector of second-order degrees can be considered models for many real complex problems. Previously, works were published in which the orders of application of invariants calculated in polynomial time were analyzed, and such variants of algorithms for which small degrees of the applied polynomial are needed. When analyzing such algorithms, there are problems of comparing the invariants under consideration — comparing by some specially selected metric that reflects the «quality» of the invariant on the subset of the set of all graphs under consideration. The article shows that when using any natural metric, the vector of second-order degrees is better than the widely used Randich index.