Let
G be an
n-vertex graph with adjacency matrix
A, and
W=[e,Ae,…,An−1e] 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−⌊n/2⌋detW is odd and square-free. In this paper, we introduce a large family of graphs
Fn={ n-vertex graphs
G:2−⌊n/2⌋detW=p2b and rank
W=n−1 over
Z/pZ}, where
b is odd and square-free,
p is an odd prime and
p∤b. We prove that any graph in
Fn 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
Fn 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
Fn 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
Fn may have a positive density (nearly
3%) and possibly almost all graphs in
Fn are DGS as
n→∞. This gives a supporting evidence for Haemers' conjecture that almost all graphs are determined by their spectra.