An arithmetic criterion for graphs being determined by their generalized $A_α$-spectrum

Let $G$ be a graph on $n$ vertices, its adjacency matrix and degree diagonal matrix are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov \cite{0007} introduced the matrix $A_α(G)=αD(G)+(1-α)A(G)$ for $α\in [0, 1].$ The $A_α$-spectrum of a graph $G$ consists of all the eigenvalues (including the multiplicities) of $A_α(G).$ A graph $G$ is said to be determined by the generalized $A_α$-spectrum (or, DGA$_α$S for short) if whenever $H$ is a graph such that $H$ and $G$ share the same $A_α$-spectrum and so do their complements, then $H$ is isomorphic to $G$. In this paper, when $α$ is rational, we present a simple arithmetic condition for a graph being DGA$_α$S. More precisely, put $A_{c_α}:={c_α}A_α(G),$ here ${c_α}$ is the smallest positive integer such that $A_{c_α}$ is an integral matrix. Let $\tilde{W}_{α}(G)=\left[{\bf 1},\frac{A_{c_α}{\bf 1}}{c_α},\ldots, \frac{A_{c_α}^{n-1}{\bf 1}}{c_α}\right]$, where ${\bf 1}$ denotes the all-ones vector. We prove that if $\frac{\det \tilde{W}_{α}(G)}{2^{\lfloor\frac{n}{2}\rfloor}}$ is an odd and square-free integer and the rank of $\tilde{W}_{α}(G)$ is full over $\mathbb{F}_p$ for each odd prime divisor $p$ of $c_α$, then $G$ is DGA$_α$S except for even $n$ and odd $c_α\,(\geqslant 3)$. By our obtained results in this paper we may deduce the main results in \cite{0005} and \cite{0002}.

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