Generalized Spectral Characterization of Mixed Graphs

The spectral characterization of graphs is an important topic in spectral graph theory, which has been studied extensively in recent years. Unlike the undirected case, however, the spectral characterization of mixed graphs (digraphs) has received much less attention so far, which will be the main focus of this paper. A mixed graph $G$ is said to be strongly determined by its generalized Hermitian spectrum (abbreviated SHDGS), if, up to isomorphism, $G$ is the unique mixed graph that is cospectral with $G$ w.r.t. the generalized Hermitian spectrum. Let $G$ be a self-converse mixed graph of order $n$ with Hermitian adjacency matrix $A$ and let $W=[e,Ae,\ldots,A^{n-1}e]$ ($e$ is the all-one vector). Suppose that $2^{-\lfloor n/2\rfloor}\det W$ is \emph{norm-free} in $\mathbb{Z}[i]$ (i.e., for any Gaussian prime $p$, the norm $N(p)=p\bar{p}$ does not divide $2^{-\lfloor n/2\rfloor}\det W$). We conjecture that every such graph is SHDGS and prove that, for any mixed graph $H$ that is cospectral with $G$ w.r.t. the generalized Hermitian spectrum, there exists a Gaussian rational unitary matrix $U$ with $Ue=e$ such that $U^*A(G)U=A(H)$ and $(1+i)U$ is a Gaussian integral matrix. We have verified the conjecture in two extremal cases when $G$ is either an undirected graph or a self-converse oriented graph. Moreover, as consequences of our main results, we prove that all directed paths of even order are SHDGS. Analogous results are also obtained in the setting of \emph{restrictive} determination by generalized Hermitian spectrum (i.e., the spectral determination within the subset of all self-converse mixed graphs), which extends a recent result of the first author on the generalized spectral characterization of undirected graphs.