刘胜久 李天瑞 谢鹏 刘佳

摘   要：传统的混合图的能量通过对方阵形式的矩陣特征值的计算而得到，难以推广应用到大规模的混合图中. 针对这个问题，本文将网络维数应用于混合图中，提出了混合图的网络能量，从而将网络能量从无向图及有向图推广应用到混合图. 混合图的网络能量可以通过混合图的节点数目及有向边与无向边的数目而得到，同时给出了混合图的网络能量的若干上下限. 在与混合图的Hermitian能量及有向图与无向图的网络能量的对比中分析了所提出的混合图的网络能量的若干重要性质，并论证了无向图、有向图及混合图的网络能量三者之间的内在关联.

关键词：图能量;网络能量;无向图;有向图;混合图

中图分类号：TP391                       文献标志码：A

Research and Application of Network Energy on Mixed Graphs

LIU Shengjiu1，2，LI Tianrui1，2?，XIE Peng1，2，LIU Jia1，2

（1. School of Information Science and Technology，Southwest Jiaotong University，Chengdu 611756，China;

2. Sichuan Key Lab of Cloud Computing and Intelligent Technique，Chengdu 611756，China）

Abstract：The energy of the traditional mixed graph is obtained by calculating the eigenvalues of the matrix in the form of a square matrix，and it is difficult to be extended to large-scale mixed graphs.  In response to this problem， this paper applies the network dimension to the mixed graph， and proposes the network energy of the mixed graph， thus the network energy is extended from the undirected graph and the directed graph to the mixed graph. The network energy of a mixed graph can be obtained by the number of nodes and the number of directed and undirected edges of the mixed graph. At the same time，several upper and lower limits of the network energy of the mixed graph are given. Comparied with the Hermitian energy of mixed graph and the network energy of directed and undirected graphs， some important properties of the proposed network energy of the mixed graph are analyzed. The internal relationships among undirected graph， directed graph and mixed graph are also demonstrated.

Key words：graph energy;network energy;undirected graph;directed graph;mixed graph

图能量最早是由Gutman于1978年正式提出的[1]，对图能量的研究可溯源到二十世纪三四十年代化学家对共轭的烃类化合物总π能量的研究. 烃类化合物总π能量可通过Huckel分子轨道理论近似求出[2-3]. 图能量定义为图的邻接矩阵所有特征值绝对值之和. 自图能量提出以来，图能量受到人们极大的关注，一系列能量，如距离能量[4]、拟Laplacian能量[5]、关联能量[6]、匹配能量[7]、Laplacian能量[8]、无符号Laplacian能量[9]、Von Neumann能量[10]、距离Laplacian能量[11]、距离无符号Laplacian能量[11]等其他类似能量，相继提出.

图能量自提出以来已在理论化学及代数图论等领域得到极为广泛的应用，一些重要研究成果相继问世. 由于大规模矩阵特征值计算极为繁琐，对图能量的研究更……