细分双冠图的拉普拉斯矩阵的广义逆及其应用

The Generalized Inverse of the Laplacian Matrix of Subdivision Double Corona with Applications

设G是一个n个顶点,m条边的连通图。在图G的每条边上添加一个新的顶点得到其细分图S(G)。G,G1和G2的细分双冠图表示为G(S)。{G1,G2},连接S(G)的第i个旧的顶点到G1的第i个复制的每一个顶点,同时连接S(G)的第j个新的顶点到G2的第j个复制的每一个顶点组成。图的拉普拉斯矩阵蕴含着图的许多结构信息。细分双冠图是一种新的图运算,同时细分双冠图的广义逆和电阻距离更加难以计算。因此,本文运用分块矩阵的广义逆理论,得到细分双冠图的拉普拉斯矩阵广义逆。作为其应用,可以求出细分双冠图中任意两点之间的电阻距离。本文的结论可以推广了已有文献的结果,可以为其他的图运算的拉普拉斯矩阵广义逆和电阻距离的计算提供借鉴和参考。

Let G be a connected graph on n vertices,medges.Let S( G) be the subdivision graph of graph G by inserting a new vertex into every edge of G. The subdivision double corona of G,G1 and G2,denoted by G(S)。{G1,G2},is the graph obtained by joining thei-th old-vertex of S( G) to every vertex of thei-th copy of G1 and thej-th new-vertex of S( G) to every vertex of thej-th copy of G2. The Laplacian matrix of the graph contains a lot of structural information. The subdivision double corona graph is a new graph operation,the computation of generalized inverse of the subdivision double corona graphs and resistance distances are more complex. Therefore, generalized inverse theory of block matrices can be applied,generalized inverse for the Laplacian matrices of subdivision double corona graphs are proposed.As its application,based on which the explicit resistance distance can be obtained for the arbitrary two-vertex in the subdivision double corona graphs. The conclusion of this paper can be extended to the results of the existing literature,and it can be used for reference for the calculation of Laplacian matrix generalized inverse and resistance distance of other graph operations.