Algorithmic methods of finite discrete structures. Automorphism of Nonseparable Graphs

TL;DR

This paper explores methods for constructing automorphism groups of finite, nonseparable graphs using spectral graph invariants and orbit concepts, providing practical examples of automorphism group construction.

Contribution

It introduces a spectral approach to automorphism group construction for nonseparable graphs, emphasizing the role of orbit and weight concepts.

Findings

Spectral invariants help identify graph automorphisms.

Orbit-based methods facilitate automorphism group construction.

Examples demonstrate the practical application of the methods.

Abstract

The monography examines the problem of constructing a group of automorphisms of a graph. A graph automorphism is a mapping of a set of vertices onto itself that preserves adjacency. The set of such automorphisms forms a vertex group of a graph or simply a graph group. The basis for constructing a group of graph automorphisms is the concept of orbit. The construction of an orbit is closely related to the quantitative assessment of a vertex or edge of a graph, called weight. To determine the weight of an element, graph invariants built on the spectrum of edge cuts and the spectrum of edge cycles are used. The weight of the graph elements allows identifying generating cycles and forming orbits. Examples are given of constructing a group of automorphisms for some types of graphs.