On the $A_{\alpha}$-spectra of some join graphs

TL;DR

This paper derives formulas for the $A_{\alpha}$-spectra of various join graphs involving regular and arbitrary graphs, enabling the construction of many $A_{\alpha}$-cospectral graph pairs.

Contribution

It provides explicit $A_{\alpha}$-spectral characterizations of several join graph constructions, extending spectral graph theory for these graph operations.

Findings

Computed $A_{\alpha}$-spectra for subdivision-vertex, subdivision-edge, $R$-vertex, and $R$-edge joins.

Established methods to generate infinitely many $A_{\alpha}$-cospectral graphs.

Extended spectral analysis to a broad class of join graphs.

Abstract

Let $G$ be a simple, connected graph and let $A(G)$ be the adjacency matrix of $G$. If $D(G)$ is the diagonal matrix of the vertex degrees of $G$, then for every real $\alpha \in [0,1]$, the matrix $A_{\alpha}(G)$ is defined as $$A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G).$$ The eigenvalues of the matrix $A_{\alpha}(G)$ form the $A_{\alpha}$-spectrum of $G$. Let $G_1 \dot{\vee} G_2$, $G_1 \underline{\vee} G_2$, $G_1 \langle \textrm{v} \rangle G_2$ and $G_1 \langle \textrm{e} \rangle G_2$ denote the subdivision-vertex join, subdivision-edge join, $R$-vertex join and $R$-edge join of two graphs $G_1$ and $G_2$, respectively. In this paper, we compute the $A_{\alpha}$-spectra of $G_1 \dot{\vee} G_2$, $G_1 \underline{\vee} G_2$, $G_1 \langle \textrm{v} \rangle G_2$ and $G_1 \langle \textrm{e} \rangle G_2$ for a regular graph $G_1$ and an arbitrary graph $G_2$ in terms of their $A_{\alpha}$-eigenvalues. As an application of these results, we construct infinitely many pairs of $A_{\alpha}$-cospectral graphs.