Spectra of subdivision-vertex and subdivision-edge neighbourhood coronae

TL;DR

This paper derives spectral properties of subdivision-vertex and subdivision-edge neighbourhood coronae graphs, enabling the construction of cospectral and expander graphs based on the spectra of original graphs.

Contribution

It provides explicit formulas for the adjacency, Laplacian, and signless Laplacian spectra of these coronae graphs in terms of the spectra of the original graphs, a novel spectral analysis approach.

Findings

Spectra of subdivision-vertex neighbourhood coronae are explicitly determined.

Spectra of subdivision-edge neighbourhood coronae are explicitly determined.

New families of cospectral and expander graphs are constructed from these results.

Abstract

Let $G=(V(G),E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The subdivision graph $\mathcal{S}(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into every edge of $G$. Let $G_1$ and $G_2$ be two vertex disjoint graphs. The subdivision-vertex neighbourhood corona of $G_1$ and $G_2$, denoted by $G_1 \boxdot G_2$, is the graph obtained from $\mathcal{S}(G_1)$ and $|V(G_1)|$ copies of $G_2$, all vertex disjoint, and joining the neighbours of the $i$th vertex of $V(G_1)$ to every vertex in the $i$th copy of $G_2$. The subdivision-edge neighbourhood corona of $G_1$ and $G_2$, denoted by $G_1 \boxminus G_2$, is the graph obtained from $\mathcal{S}(G_1)$ and $|I(G_1)|$ copies of $G_2$, all vertex disjoint, and joining the neighbours of the $i$th vertex of $I(G_1)$ to every vertex in the $i$th copy of $G_2$, where $I(G_1)$ is the set of inserted vertices of $\mathcal{S}(G_1)$. In this paper we determine the adjacency spectra, the Laplacian spectra and the signless Laplacian spectra of $G_1\boxdot G_2$ (respectively, $G_1\boxminus G_2$) in terms of the corresponding spectra of $G_1$ and $G_2$. As applications, these results enable us to construct infinitely many pairs of cospectral graphs, and using the results on the Laplacian spectra of subdivision-vertex neighbourhood coronae, new families of expander graphs are constructed from known ones.