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.

Some Graphs Determined by their Signless Laplacian (Distance) Spectra

In literature, there are some results known about spectral determination of graphs with many edges. In [M.~C\'{a}mara and W.H.~Haemers. Spectral characterizations of almost complete graphs. {\em Discrete Appl. Math.}, 176:19--23, 2014.], C\'amara and Haemers studied complete graph with some edges deleted for spectral determination. In fact, they found that if the deleted edges form a matching, a complete graph $K_m$ provided $m \le n-2$, or a complete bipartite graph, then it is determined by its adjacency spectrum. In this paper, the graph $K_{n}\backslash K_{l,m}$ $(n>l+m)$ which is obtained from the complete graph $K_{n}$ by removing all the edges of a complete bipartite subgraph $K_{l,m}$ is studied. It is shown that the graph $K_{n}\backslash K_{1,m}$ with $m\ge4$ is determined by its signless Laplacian spectrum, and it is proved that the graph $K_{n}\backslash K_{l,m}$ is determined by its distance spectrum. The signless Laplacian spectral determination of the multicone graph $K_{n-2\alpha}\vee \alpha K_{2}$ was studied by Bu and Zhou in [C.~Bu and J.~Zhou. Signless Laplacian spectral characterization of the cones over some regular graphs. {\em Linear Algebra Appl.}, 436:3634--3641, 2012.] and Xu and He in [L. Xu and C. He. On the signless Laplacian spectral determination of the join of regular graphs. {\em Discrete Math. Algorithm. Appl.}, 6:1450050, 2014.] only for $n-2\alpha=1 ~\text{or}~ 2$. Here, this problem is completely solved for all positive integer $n-2\alpha$. The proposed approach is entirely different from those given by Bu and Zhou, and Xu and He.