Cospectral graphs obtained by edge deletion

TL;DR

This paper investigates how deleting edges in 1-walk-regular graphs can produce new cospectral graphs across various matrix types, revealing new relationships between graph modifications and spectral properties.

Contribution

It introduces methods to generate cospectral graphs through edge deletion in 1-walk-regular graphs, expanding understanding of spectral graph theory and graph isomorphism.

Findings

Edge deletion in 1-walk-regular graphs yields cospectral graphs.

Cospectrality is preserved under certain edge deletions in cliques.

Results apply to adjacency, Laplacian, unsigned Laplacian, and normalized Laplacian matrices.

Abstract

Let $M\circ N$ denote the Schur product of two matrices $M$ and $N$. A graph $X$ with adjacency matrix $A$ is walk regular if $A^k\circ I$ is a constant times $I$ for each $k\ge0$, and $X$ is 1-walk-regular if it is walk regular and $A^k\circ A$ is a constant times $A$ for each $k\ge0$. Assume $X$ is 1-walk regular. Here we show that by deleting an edge in $X$, or deleting edges of a graph inside a clique of $X$, we obtain families of graphs that are not necessarily isomorphic, but are cospectral with respect to four types of matrices: the adjacency matrix, Laplacian matrix, unsigned Laplacian matrix, and normalized Laplacian matrix. Furthermore, we show that removing edges of Laplacian cospectral graphs in cliques of a 1-walk regular graph results in Laplacian cospectral graphs; removing edges of unsigned Laplacian cospectral graphs whose complements are also cospectral with respect to the unsigned Laplacian in cliques of a 1-walk regular graph results in unsigned Laplacian cospectral graphs.