New constructions of non-regular cospectral graphs

TL;DR

This paper introduces new methods for constructing non-regular, non-isomorphic graphs that are cospectral with respect to multiple matrices by analyzing specific graph joins and their spectra.

Contribution

It provides explicit formulas for the spectra of certain graph joins and uses these to generate new classes of cospectral non-regular graphs.

Findings

Derived formulas for adjacency, Laplacian, signless Laplacian, and normalized Laplacian spectra of graph joins.

Constructed non-regular, non-isomorphic cospectral graphs for multiple matrices.

Extended spectral analysis to non-regular graphs using join operations.

Abstract

We consider two types of joins of graphs $G_{1}$ and $G_{2}$, $G_{1}\veebar G_{2}$ - the Neighbors Splitting Join and $G_{1}\underset{=}{\lor}G_{2}$ - the Non Neighbors Splitting Join, and compute the adjacency characteristic polynomial, the Laplacian characteristic polynomial and the signless Laplacian characteristic polynomial of these joins. When $G_{1}$ and $G_{2}$ are regular, we compute the adjacency spectrum, the Laplacian spectrum, the signless Laplacian spectrum of $G_{1}\underset{=}{\lor}G_{2}$ and the normalized Laplacian spectrum of $G_{1}\veebar G_{2}$ and $G_{1}\underset{=}{\lor}G_{2}$. We use these results to construct non regular, non isomorphic graphs that are cospectral with respect to the four matrices: adjacency, Laplacian , signless Laplacian and normalized Laplacian.