Hermitian adjacency matrix of the second kind for mixed graphs

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. We identify all mixed graphs whose eigenvalues of its N-matrix lie in the range (-alpha, alpha) for alpha epsilon[root 2, root 3, root 2 ] (c) 2022 Elsevier B.V. All rights reserved.