3-Regular mixed graphs with optimum Hermitian energy

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• 作者：Xiaolin Chen ^{chxlnk@163.com" class="auth_mail" title="E-mail the corresponding author} ; Xueliang Li ; ^{lxl@nankai.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Yingying Zhang ^{zyydlwyx@163.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05C20 ; 05C50 ; 05C90
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：1 May 2016
• 年：2016
• 卷：496
• 期：Complete
• 页码：475-486
• 全文大小：371 K

文摘

Let G   be a simple undirected graph, and G^ϕ$G^{\varphi }$ be a mixed graph of G with the generalized orientation ϕ   and Hermitian-adjacency matrix H(G^ϕ)$H(G^{\varphi })$. Then G   is called the underlying graph of G^ϕ$G^{\varphi }$. The Hermitian energy of the mixed graph G^ϕ$G^{\varphi }$, denoted by E_H(G^ϕ)${\mathcal{E}}_H(G^{\varphi })$, is defined as the sum of all the singular values of H(G^ϕ)$H(G^{\varphi })$. A k-regular mixed graph on n   vertices having Hermitian energy $n\sqrt{k}$ 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=3$k=3$.