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A graph $G$ is said to be determined by its generalized spectrum (DGS forshort) if for any graph $H$, $H$ and $G$ are cospectral with cospectralcomplements implies that $H$ is isomorphic to $G$.  It turns out that whether a graph $G$ is DGS is closely related to thearithmetic properties of its walk-matrix. More precisely, let $A$ be theadjacency 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, assuggested 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 asimple arithmetic condition for determining whether a graph is DGS.

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