$3$-Regular mixed graphs with optimum Hermitian energy

Abstract

Let $G$ be a simple undirected graph, and $G^\phi$ be a mixed graph of $G$ with the generalized orientation $\phi$ and Hermitian-adjacency matrix $H(G^\phi)$. Then $G$ is called the underlying graph of $G^\phi$. The Hermitian energy of the mixed graph $G^\phi$, denoted by $\mathcal{E}_H(G^\phi)$, is defined as the sum of all the singular values of $H(G^\phi)$. A $k$-regular mixed graph on $n$ vertices having Hermitian energy $n\sqrt{k}$ is called a $k$-regular optimum Hermitian energy mixed graph. In this paper, we first focus on the problem proposed by Liu and Li [J. Liu, X. Li, Hermitian-adjacency matrices and Hermitian energies of mixed graphs, Linear Algebra Appl. 466(2015), 182--207] of determining all the $3$-regular connected optimum Hermitian energy mixed graphs. We then prove that optimum Hermitian energy oriented graphs with underlying graph hypercube are unique (up to switching equivalence).