On the characteristic polynomial and energy of Hermitian quasi-Laplacian matrix of mixed graphs

Abstract

A mixed graph is a graph whose edge set consists of both oriented and unoriented edges. The Hermitian-adjacency matrix of an [Formula: see text]-vertex mixed graph is a square matrix [Formula: see text] of order [Formula: see text], where [Formula: see text] if there is an arc from [Formula: see text] to [Formula: see text] and [Formula: see text] if there is an edge between [Formula: see text] and [Formula: see text], and [Formula: see text] otherwise. Let [Formula: see text] be a diagonal matrix, where [Formula: see text] is the degree of [Formula: see text] in the underlying graph of [Formula: see text]. The matrices [Formula: see text] and [Formula: see text] are, respectively, the Hermitian Laplacian and Hermitian quasi-Laplacian matrix of the mixed graph [Formula: see text]. In this paper, we first found coefficients of the characteristic polynomial of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph [Formula: see text]. Second, we discussed relationship between the spectra of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph [Formula: see text].