Hermitian adjacency matrix of the second kind for mixed graphs

作者：Li, Shuchao^*; Yu, Yuantian

来源：Discrete Mathematics, 2022, 345(5): 112798.

DOI：10.1016/j.disc.2022.112798

• Mixed graph
• Spectral radius
• Characteristic polynomial
• Switching equivalence
• Cospectrality
• Rank

This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind (N-matrix for short) introduced by Mohar [25]. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc V from u to v is equal to the sixth root of unity omega = 1+i root 3/ 2 (and its symmetric entry is (omega) over bar = 1-1 root 3/2); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. The 3 main results of this paper include the following: equivalent conditions for a mixed graph that shares the same spectrum of its N-matrix with its underlying graph are given. A sharp upper bound on the spectral radius is established and the corresponding extremal mixed graphs are identified. Operations which are called two-way and three-way switchings are discussed-they give rise to some cospectral mixed graphs. We extract all the mixed graphs whose rank of its N-matrix is 2 (resp. 3). Furthermore, we show that if M-G is a connected mixed graph with rank 2, then M-G is switching equivalent to each connected mixed graph to which it is cospectral. However, this does not hold for some connected mixed graphs with rank 3.