Constructing Families of Cospectral Regular Graphs

TL;DR

This paper presents a straightforward method for creating infinite families of cospectral regular graphs, especially cubic graphs, demonstrating that many such graphs share the same spectral properties.

Contribution

The paper introduces a new simple construction technique for generating infinite cospectral regular graphs, extending Schwenk's property for specific cases.

Findings

The construction produces a large proportion of cubic graphs that are cospectral with others.

Computational results confirm the effectiveness of the method for cubic graphs.

The method applies to special cases of Schwenk's property.

Abstract

A set of graphs are called cospectral if their adjacency matrices have the same characteristic polynomial. In this paper we introduce a simple method for constructing infinite families of cospectral regular graphs. The construction is valid for special cases of a property introduced by Schwenk. For the case of cubic (3-regular) graphs, computational results are given which show that the construction generates a large proportion of the cubic graphs, which are cospectral with another cubic graph.