The k-generalized Hermitian adjacency matrices for mixed graphs

Abstract

This article gives some fundamental introduction to spectra of mixed graphs via its k-generalized Hermitian adjacency matrix. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from u to v is equal to the kth root of unity e2πik (and its symmetric entry is e−2πik); 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,e2πik,e−2πik}, 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 (Hk-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 Hk-rank 2 and 3, respectively.