The Hermitian permanental polynomial of a mixed graph

Dougaicuomao Ji1, Tingzeng Wu1,2, Shunyi Liu3
1School of Mathematics and Statistics, Qinghai Minzu University, Xining, Qinghai 810007, P.R.~China
2Qinghai Institute of Applied Mathematics, Xining, Qinghai 810007, P.R. China
3School of Science, Chang’an University, Xi’an, Shaanxi 710064, P.R. China

Abstract

Let \(G\) be a simple connected mixed graph, and let \(H(G)\) denote the Hermitian adjacency matrix of \(G\). The Hermitian permanental polynomial of \(G\) is defined as \(\pi(G; x) = \operatorname{per}(xI – H(G))\), where \(\operatorname{per}(\cdot)\) represents the permanent and \(I\) is the identity matrix. In this paper, we first derive fundamental properties of the Hermitian permanental polynomial for mixed graphs and establish explicit formulas relating its coefficients to those of the characteristic polynomial. We then analyze the root distribution of this polynomial, determining the number of zero roots for several special classes of mixed graphs. Finally, we characterize mixed graphs that remain cospectral under four‑way switching and prove that this operation preserves the permanental spectrum.

Keywords: mixed graph, Hermitian adjacency matrix, Hermitian permanental polynomial