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

School of Mathematical Sciences, Anhui University, Hefei 230601, P. R. China

and

Basic Science Department, College of Science and Theoretical Studies, Saudi Electronic University, Abha-Male, Saudi Arabia

E-mail Address: a.ganie@seu.edu.sa

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https://doi.org/10.1142/S1793557123501164Cited by:0 (Source: Crossref)

Abstract

A mixed graph is a graph whose edge set consists of both oriented and unoriented edges. The Hermitian-adjacency matrix of an $n$ $n$ -vertex mixed graph is a square matrix $H (M) = [h_{j k}]$ $H (M) = [h_{j k}]$ of order $n$ $n$ , where $h_{j k} = ˙ ι = - h_{k j}$ $h_{j k} = \dot{ι} = - h_{k j}$ if there is an arc from $v_{j}$ $v_{j}$ to $v_{k}$ $v_{k}$ and $h_{j k} = 1$ $h_{j k} = 1$ if there is an edge between $v_{j}$ $v_{j}$ and $v_{k}$ $v_{k}$ , and $h_{j k} = 0$ $h_{j k} = 0$ otherwise. Let $D (M) = [d_{j j}]$ $D (M) = [d_{j j}]$ be a diagonal matrix, where $d_{j j}$ $d_{j j}$ is the degree of $v_{j}$ $v_{j}$ in the underlying graph of $M$ $M$ . The matrices $L (M) = D (M) - H (M)$ $L (M) = D (M) - H (M)$ and $Q (M) = D (M) + H (M)$ $Q (M) = D (M) + H (M)$ are, respectively, the Hermitian Laplacian and Hermitian quasi-Laplacian matrix of the mixed graph $M$ $M$ . In this paper, we first found coefficients of the characteristic polynomial of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph $M$ $M$ . Second, we discussed relationship between the spectra of Hermitian Laplacian and Hermitian quasi-Laplacian matrices of the mixed graph $M$ $M$ .

Communicated by I. Peterin

Keywords:

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