2014/12/11 by Chen, Xiaolin, Li, Xueliang, Zhang, Yingying
#05C20 #05C50 #05C90 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1412.3669
Let G be a simple undirected graph, and Gϕ be a mixed graph of G with the generalized orientation ϕ and Hermitian-adjacency matrix H(Gϕ). Then G is called the underlying graph of Gϕ. The Hermitian energy of the mixed graph Gϕ, denoted by EH(Gϕ), is defined as the sum of all the singular values of H(Gϕ). A k-regular mixed graph on n vertices having Hermitian energy n√(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).