3-Regular mixed graphs with optimum Hermitian energy

Let G   be a simple undirected graph, and GϕGϕ be a mixed graph of G with the generalized orientation ϕ   and Hermitian-adjacency matrix H(Gϕ)H(Gϕ). Then G   is called the underlying graph of GϕGϕ. The Hermitian energy of the mixed graph GϕGϕ, denoted by EH(Gϕ)EH(Gϕ), is defined as the sum of all the singular values of H(Gϕ)H(Gϕ). A k-regular mixed graph on n   vertices having Hermitian energy nk is called a k-regular optimum Hermitian energy mixed graph. Liu and Li (2015) [11] proposed the problem of determining all the k  -regular connected optimum Hermitian energy mixed graphs. This paper is to give a solution to the problem for the case k=3k=3.