Lu Yong, Wang Ligong, Zhou Qiannan
Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi'an, Shaanxi 710072 People's Republic of China.
J Inequal Appl. 2017;2017(1):54. doi: 10.1186/s13660-017-1329-8. Epub 2017 Mar 3.
Let be a mixed graph and [Formula: see text] be its Hermitian-adjacency matrix. If we add a Randić weight to every edge and arc in , then we can get a new weighted Hermitian-adjacency matrix. What are the properties of this new matrix? Motivated by this, we define the Hermitian-Randić matrix [Formula: see text] of a mixed graph , where [Formula: see text] ([Formula: see text]) if [Formula: see text] is an arc of , [Formula: see text] if [Formula: see text] is an undirected edge of , and [Formula: see text] otherwise. In this paper, firstly, we compute the characteristic polynomial of the Hermitian-Randić matrix of a mixed graph. Furthermore, we give bounds on the Hermitian-Randić energy of a general mixed graph. Finally, we give some results about the Hermitian-Randić energy of mixed trees.
设(G)为一个混合图，(H(G))为其埃尔米特邻接矩阵。若我们给(G)中的每条边和弧都添加一个兰迪奇权重，那么我们就能得到一个新的加权埃尔米特邻接矩阵。这个新矩阵有哪些性质呢？受此启发，我们定义混合图(G)的埃尔米特 - 兰迪奇矩阵(R(G))，其中若(uv)是(G)的一条弧，则(R_{uv}=\frac{1}{\sqrt{d_{u}d_{v}}})（(d_{u})和(d_{v})分别是顶点(u)和(v)的度），若(uv)是(G)的一条无向边，则(R_{uv}=\frac{1}{\sqrt{d_{u}d_{v}}})，否则(R_{uv}=0)。在本文中，首先，我们计算混合图的埃尔米特 - 兰迪奇矩阵的特征多项式。此外，我们给出一般混合图的埃尔米特 - 兰迪奇能量的界。最后，我们给出一些关于混合树的埃尔米特 - 兰迪奇能量的结果。
