Further properties of Hermitian adjacency matrix of mixed graphs

Abstract

The Hermitian adjacency matrix of a mixed graph extends the classical adjacency matrix to graphs containing both edges and arcs. In this paper, we establish a unified generalised interpretation for the powers of the Hermitian adjacency matrix, where arcs may be traversed in either direction and the orientation is encoded through complex conjugate weights. We show that each entry of the kth power of the Hermitian adjacency matrix corresponds to a weighted sum of all generalised walks of length k between the corresponding vertices. Using this correspondence, we derive explicit combinatorial expansions for the determinant and for the coefficients of the characteristic polynomial in terms of spanning sub-mixed graphs. Applications to path and cycle digraphs are presented, and several spectral consequences are discussed.