Cospectral constructions for several graph matrices using cousin vertices

Kate Lorenzen

doi:10.1515/spma-2020-0143

Abstract

Abstract Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us establish patterns about structural information not preserved by the spectrum. We generalize a construction for cospectral graphs previously given for the distance Laplacian matrix to a larger family of graphs. In addition, we show that with appropriate assumptions this generalized construction extends to the adjacency matrix, combinatorial Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and distance matrix. We conclude by enumerating the prevelance of this construction in small graphs for the adjacency matrix, combinatorial Laplacian matrix, and distance Laplacian matrix.

Highlights

In the most general terms, graph theory is the study of objects and their relations
We have presented an extension of a construction method and applied it to many matrices
A natural question about this construction method is what fraction of cospectral graphs does this explain for each matrix as the number of vertices increases? In addition, there are smaller examples those shown in this paper for some matrices, but it is unknown if there is a smaller example for the distance matrix

Summary

Introduction

In the most general terms, graph theory is the study of objects (vertices) and their relations (edges). For the combinatorial Laplacian matrix, the construction says that if we have two sets of isolated twin vertices of the same size and they have the same degree, the pair of graphs produced by adding k edges into one set or the other are cospectral. We will show that the twin condition can be relaxed for the combinatorial Laplacian matrix when constructing cospectral graphs and can be broadened to include all adjacency and distance matrices.

Results

Conclusion