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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 $Gcolon, 2^{-lfloor n/2 rfloor}det W =p^2b$ and rank$W=n-1$ over $mathbb{Z}/pmathbb{Z}},$ where $b$ is odd and square-free, $p$ is an odd prime and $p mid b$. 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 $nrightarrow infty$. This gives a supporting evidence for Haemers conjecture that almost all graphs are determined by their spectra.
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We prove an upper bound on the number of pairwise strongly cospectral vertices in a normal Cayley graph, in terms of the multiplicities of its eigenvalues. We use this to determine an explicit bound in Cayley graphs of $mathbb{Z}_2^d$ and $mathbb{Z}_4^d$. We also provide some infinite families of Cayley graphs of $mathbb{Z}_2^d$ with a set of four pairwise strongly cospectral vertices and show that such graphs exist in every dimension.
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Given a graph, we can form a spanning forest by first sorting the edges in some order, and then only keep edges incident to a vertex which is not incident to any previous edge. The resulting forest is dependent on the ordering of the edges, and so we can ask, for example, how likely is it for the process to produce a graph with $k$ trees. We look at all graphs which can produce at most two trees in this process and determine the probabilities of having either one or two trees. From this we construct infinite families of graphs which are non-isomorphic but produce the same probabilities.
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