  Let $G$ be an $n$-vertex graph with adjacency matrix $A$, and
$W=[e,Ae,\ldots,A^{n-1}e]$ be the walk matrix of $G$, where $e$ is the all-one
vector. In Wang [J. Combin. Theory, Ser. B, 122 (2017): 438-451], the author
showed that any graph $G$ is uniquely determined by its generalized spectrum
(DGS) whenever $2^{-\lfloor n/2 \rfloor}\det W$ is odd and square-free. In this
paper, we introduce a large family of graphs $\mathcal{F}_n=\{$ $n$-vertex
graphs $G\colon\, 2^{-\lfloor n/2 \rfloor}\det W =p^2b$ and rank$W=n-1$ over
$\mathbb{Z}/p\mathbb{Z}\},$ where $b$ is odd and square-free, $p$ is an odd
prime and $p\nmid b$. We prove that any graph in $\mathcal{F}_n$ either is DGS
or has exactly one generalized cospectral mate up to isomorphism. Moreover, we
show that the problem of finding the generalized cospectral mate for a graph in
$\mathcal{F}_n$ is equivalent to that of generating an appropriate rational
orthogonal matrix from a given integral vector. This equivalence essentially
depends on an amazing property of graphs in terms of generalized spectra, which
states that any symmetric integral matrix generalized cospectral with the
adjacency matrix of some graph must be an adjacency matrix. Based on this
equivalence, we develop an efficient algorithm to decide whether a given graph
in $\mathcal{F}_n$ is DGS and further to find the unique generalized cospectral
mate when it is not. We give some experimental results on graphs with at most
20 vertices, which suggest that $\mathcal{F}_n$ may have a positive density
(nearly $3\%$) and possibly almost all graphs in $\mathcal{F}_n$ are DGS as
$n\rightarrow \infty$. This gives a supporting evidence for Haemers' conjecture
that almost all graphs are determined by their spectra.
