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