The k-generalized Hermitian adjacency matrices for mixed graphs

This article gives some fundamental introduction to spectra of mixed graphs via its kgeneralized Hermitian adjacency matrix. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from uto vis equal to the kth root of unity e(-2pi/k) (and its symmetric entry is e(-2pi/k)); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. For all positive integers k, the non-zero entries of the above matrix are chosen from the gain set {1, e(2pi/k), e(-2pi/k)}, which is not closed under multiplication when k >= 4. In this paper, for all positive integers k, we extract all the mixed graphs whose k-generalized Hermitian adjacency rank (H-k-rank for short) is 3, which partially answers a question proposed by Wissing and van Dam [34]. Furthermore, we study the spectral determination of mixed graphs with H-k-rank 2 and 3, respectively. (C) 2022 Elsevier B.V. All rights reserved.