Hermitian-adjacency matrices and Hermitian energies of mixed graphs

Abstract

A complex adjacency matrix of a mixed graph is introduced in the present paper, which is a Hermitian matrix and called the Hermitian-adjacency matrix. It incorporates both adjacency matrix of an undirected graph and skew-adjacency matrix of an oriented graph. Some of its properties are studied. Furthermore, properties of its characteristic polynomial are studied. Cospectral problems among mixed graphs, including mixed graphs and their underlying graphs, oriented graphs and their underlying graphs, are studied. We give equivalent conditions for a mixed graph (especially oriented graph) that share the same spectrum with its underlying graph. As a consequence, we reconfirm a conjecture which was proposed by Cui and Hou in Ref. [8]. We also show that the spectrum of the Hermitian matrix of a mixed graph is invariant when changing the value of any its cut edge (if any). Correspondingly, an energy of a mixed graph is introduced and called the Hermitian energy. It incorporates both the energy of an undirected graph and the skew energy of an oriented graph. Some of its bounds are given. Especially, the mixed graphs with optimal upper bound of Hermitian energy are characterized. An infinite family of mixed graphs attaining the maximum Hermitian energy is constructed. Moreover, the Hermitian energy of a mixed tree is showed to be equal to ✩ 183 the energy of its underlying tree. Finally, the integral formula for Hermitian energy of a mixed graph is given.

Hermitian Adjacency Matrix of Digraphs and Mixed Graphs

“…In particular, the expression for from Theorem yields the result in Corollary since all basic graphs of order 2 are single edges. Using this theorem, we also obtain the following corollary, which has also been used in . Corollary Suppose that is a digon in X .…”

“…In this article, we investigate basic properties of the Hermitian adjacency matrix, complementary to the results of Liu and Li . We observe that the largest eigenvalue of the Hermitian adjacency matrix is upper bounded by the maximum degree of the underlying graph and we characterize the case when equality is attained (see Theorem ).…”

Hermitian Laplacian matrix and positive of mixed graphs

“…Since each cycle of the subgraph is a cycle of the positive mixed graph, the following is immediate. 1 and V 2 such that every edge within V 1 or V 2 is undirected, every edge between V 1 and V 2 is directed and all orientations are from V 1 to V 2 or from V 2 to V 1 .…”

“…This matrix was introduced by Liu and Li [1] and independently by Guo and Mohar [2]. We will denote by λ i (M) the jth largest eigenvalue of H(M) (multiplicities counted).…”

“…Liu and Li [1] studied some properties of Hermitian adjacency matrix. Furthermore, they investigated the cospectral problems among mixed graphs, including mixed graphs and their underlying graphs, oriented graphs and their underlying graphs.…”