Spectra of subdivision-vertex join and subdivision-edge join of two graphs

TL;DR

This paper derives spectral properties and structural metrics for subdivision-vertex and subdivision-edge joins of graphs, providing formulas based on original graphs' spectra and enabling the construction of cospectral graphs.

Contribution

It introduces explicit spectral formulas for subdivision-vertex and subdivision-edge joins, expanding understanding of graph spectra and structural invariants.

Findings

Spectra of the joins are expressed in terms of original graphs' spectra.

Infinitely many pairs of cospectral graphs are constructed.

Number of spanning trees and Kirchhoff index are determined for the joins.

Abstract

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 \emph{subdivision-vertex join} of $G_1$ and $G_2$, denoted by $G_1\dot{\vee}G_2$, is the graph obtained from $\mathcal{S}(G_1)$ and $G_2$ by joining every vertex of $V(G_1)$ with every vertex of $V(G_2)$. The \emph{subdivision-edge join} of $G_1$ and $G_2$, denoted by $G_1\underline{\vee}G_2$, is the graph obtained from $\mathcal{S}(G_1)$ and $G_2$ by joining every vertex of $I(G_1)$ with every vertex of $V(G_2)$, 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 $G_1\dot{\vee}G_2$ (respectively, $G_1\underline{\vee}G_2$) for a regular graph $G_1$ and an arbitrary graph $G_2$, 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. We also give the number of the spanning trees and the Kirchhoff index of $G_1\dot{\vee}G_2$ (respectively, $G_1\underline{\vee}G_2$) for a regular graph $G_1$ and an arbitrary graph $G_2$.