Operators and Graph Isomorphism

Abstract

The article considers a method for recognizing graph isomorphism. The basis of the method is the construction of a chain of transformations of the adjacency matrix of the edge graph L(G) using the properties of the nilpotent operator of the subgraph space. Traditional approaches to graph structure analysis are basically grounded on the use of the adjacency matrix A(G) of the graph. In this article, we propose an approach based on the use of the adjacency matrix of the edge graph A(L(G)) considered as a linear operator of the subgraph space of the graph G. The spectrum of elements of the subgraph space of the graph characterizing the set of edge graph cuts. 7he concept of an edge cut of a graph is introduced and described in the article. It is shown that the formation of a chain of edge cuts generates a spectrum of graph edge cuts, which, in turn, determines the metric properties of the edges and vertices of the graph. The metric properties of the spectrum of edge cuts allow constructing numerical characteristics of the graph structure and the invariant of graph edge cuts, as well as applying it to the solution of the graph isomorphism recognition problem. In this paper, we consider the main algorithmic properties of the spectrum of graph edge cuts and propose an approach for constructing an invariant of edge cuts. An algorithm for determining the invariant of graph edge cuts based on the concept of the spectrum of edge cuts is proposed. It is shown that the construction of an invariant of edge cuts of a graph belongs to the class of polynomial problems.