  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.
