This paper is concerned with the identification of important nodes in node-weighted graphs by applying matrix functions, in particular the matrix exponential. Many tools that use an adjacency matrix for a graph have been developed to study the importance of the nodes in unweighted or edge-weighted networks. However, adjacency matrices for node-weighted graphs have not received much attention. The present paper proposes using a line graph associated with a node-weighted graph to construct an edge-weighted graph that can be analyzed with available methods. Both undirected and directed graphs with positive node weights are considered. We show that when the weight of a node increases, the importance of this node in the graph increases as well, provided that the adjacency matrix is suitably scaled. Applications to real-life problems are presented.

本文涉及通过应用矩阵函数（尤其是矩阵指数）在节点加权图中重要节点的识别。已经开发出了许多用于图表的邻接矩阵的工具，以研究未加权或边缘加权网络中节点的重要性。但是，节点加权图的邻接矩阵并未引起太多关注。本文提出使用与节点加权图相关联的线图来构建可以使用可用方法进行分析的边缘加权图。带有正节点权重的无向图和有向图都将被考虑。我们显示，当节点的权重增加时，只要适当地缩放了邻接矩阵，该节点在图中的重要性也将增加。介绍了对现实生活中的问题的应用。