Mixed Graphs Determined by their Generalized Hermitian Adjacency Spectrum Based on Eisenstein Integers

A mixed graph is a graph obtained from a simple undirected graph by orientating a subset of edges. In 2020, Mohar introduced a new kind of Hermitian adjacency matrix (called Eisenstein adjacency matrix) of a mixed graph using a primitive sixth root of unity, which has some advantages over the one proposed by Guo and Mohar in 2017, and independently by Liu and Li in 2015 (called Gaussian adjacency matrix). We consider the problem of generalized spectral characterizations of mixed graphs based on the Eisenstein adjacency matrix. A simple sufficient condition is given for a self-converse mixed graph to be determined by its generalized Eisenstein spectrum based on the ring of Eisenstein integers. Numerical experiments are also presented which show that the generalized Eisenstein spectrum is superior to the generalized Gaussian spectrum in distinguishing mixed graphs.