$3$-Regular mixed graphs with optimum Hermitian energy
  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). 
Xiaolin Chen, Xueliang Li, Yingying Zhang
math.CO