Algorithmic methods of finite discrete structures. Isomorphism of Nonseparable Graphs

TL;DR

This paper introduces a novel approach to graph isomorphism using spectra of edge cuts and cycles, based on incidence matrices, and proves their sufficiency for determining graph isomorphism.

Contribution

It proposes a new spectral method for graph isomorphism based on edge cuts and cycles, extending previous structural analysis techniques.

Findings

Spectra of edge cuts and cycles determine graph isomorphism.

Construction of spectra is independent of vertex and edge numbering.

The method relates to Whitney's theorem on graph structure.

Abstract

In this monography, it is proposed to consider the concepts of spectra of edge cuts and edge cycles of a graph as a basic mathematical structure for solving the problem of graph isomorphism. An edge cut is defined by an edge and the vertices incident to it. In contrast to the generation of iterated edge graphs, we consider an iterated chain of qualicuts of the original graph, generated by edge cuts and determined by a recurrence relation. An edge cycle is defined by the set of isometric cycles of a graph. The monography examines the issues of constructing the spectrum of edge cuts Ws and the spectrum of edge cycles Tc of a graph G. It is shown that the formation of spectra is based on the incidence matrix of the graph. The independence of the construction of the graph structure from the numbering of vertices and edges is shown. The necessity and sufficiency of the spectra of edge cuts and the spectrum of edge cycles for determining the isomorphism of graph structures is shown. The relation between the internal structures of the graph and Whitney's theorem is considered.