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A simple arithmetic criterion for graphs being determined by their generalized spectra

A graph $G$ is said to be determined by its generalized spectrum (DGS for short) if for any graph $H$, $H$ and $G$ are cospectral with cospectral complements implies that $H$ is isomorphic to $G$. It turns out that whether a graph $G$ is DGS is closely related to the arithmetic properties of its walk-matrix. More precisely, let $A$ be the adjacency matrix of a graph $G$, and let $W =[e, Ae, A^2e,...,A^{n-1}e]$ ($e$ is the all-one vector) be its \textit{walk-matrix}. Denote by $\mathcal{G}_n$ the set of all graphs on $n$ vertices with $\det(W)\neq 0$. In [Wang, Generalized spectral characterization of graphs revisited, The Electronic J. Combin., 20 (4),(2013), #$P_4$], the author defined a large family of graphs $$\mathcal{F}_n = \{G \in{\mathcal{G}_n}|\frac{\det(W)}{2^{\lfloor\frac{n}{2}\rfloor}}{is~ an ~odd~ square-free~ integer}\}$$ (which may have positive density among all graphs, as suggested by some numerical experiments) and conjectured every graph in $\mathcal{F}_n$ is DGS. In this paper, we show that the conjecture is actually true, thereby giving a simple arithmetic condition for determining whether a graph is DGS.