Graphs determined by their $A_{\alpha}$-spectra

TL;DR

This paper investigates which graphs are uniquely identified by their $A_{\alpha}$-spectra, introducing new results for various graph classes and conditions, and explores the spectral determination of graph joins for $\alpha$ in (1/2, 1).

Contribution

It proves that several classes of graphs are determined by their $A_{\alpha}$-spectra for $0 \leq \alpha < 1$, and establishes conditions linking spectral determination of graphs and their joins.

Findings

Complete graphs, stars, paths, and certain cycle unions are determined by their $A_{\alpha}$-spectra.

For regular graphs, spectral determination of $G$ is equivalent to that of $G \vee K_m$ for $\frac{1}{2}<\alpha<1$.

The join $K_m \vee P_n$ is determined by its $A_{\alpha}$-spectrum for $\frac{1}{2}<\alpha<1$.

Abstract

Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G) $$ for any real $\alpha\in [0,1]$. The collection of eigenvalues of $A_{\alpha}(G)$ together with multiplicities are called the \emph{$A_{\alpha}$-spectrum} of $G$. A graph $G$ is said to be \emph{determined by its $A_{\alpha}$-spectrum} if all graphs having the same $A_{\alpha}$-spectrum as $G$ are isomorphic to $G$. We first prove that some graphs are determined by its $A_{\alpha}$-spectrum for $0\leq\alpha<1$, including the complete graph $K_m$, the star $K_{1,n-1}$, the path $P_n$, the union of cycles and the complement of the union of cycles, the union of $K_2$ and $K_1$ and the complement of the union of $K_2$ and $K_1$, and the complement of $P_n$. Setting $\alpha=0$ or $\frac{1}{2}$, those graphs are determined by $A$- or $Q$-spectra. Secondly, when $G$ is regular, we show that $G$ is determined by its $A_{\alpha}$-spectrum if and only if the join $G\vee K_m$ is determined by its $A_{\alpha}$-spectrum for $\frac{1}{2}<\alpha<1$. Furthermore, we also show that the join $K_m\vee P_n$ is determined by its $A_{\alpha}$-spectrum for $\frac{1}{2}<\alpha<1$. In the end, we pose some related open problems for future study.