SIAM J. MATRIX ANAL. APPL. c 1997 Society for Industrial and Applied Mathematics Vol. 18, No. 4, pp. 827–841, October 1997 003
DISTANCESINWEIGHTEDTREESANDGROUPINVERSEOF LAPLACIAN MATRICES∗
STEPHEN J. KIRKLAND† , MICHAEL NEUMANN‡ , AND BRYAN L. SHADER§
Abstract. In this paper we find formulas for group inverses of Laplacians of weighted trees. We then develop a relationship between entries of the group inverse and various distance functions on trees. In particular, we show that the maximal and minimal entries on the diagonal of the group inverse correspond to certain pendant vertices of the tree and to a centroid of the tree, respectively. We also give a characterization for the group inverses of the Laplacian of an unweighted tree to be an M-matrix.
Key words. Laplacian matrix, generalized inverse, weighted tree
AMS subject classifications. Primary, 15A09; Secondary, 05C50
PII. S0895479896298713
1. Introduction. In this paper, for a weighted tree on n-vertices, we bring into focus the relationship between the inverse weighted generalized distance function whichisdefinedonitsverticesandtheentriesofthegroupinverseoftheLaplacian matrix associated with the tree. In so doing we generalize and extend results which we have obtained in a previous paper for unweighted trees. We also shift the frame of reference for what we termed in that paper as bottleneck numbers for the tree to the inverse weighted distance function. Other results follow. An undirected weighted graph on n vertices is a graph, G, each of whose edges e has been labeled by a positive real number, w(e), which is called the weight of the edge e. Taking the vertices of G to be 1, 2,...,n,theLaplacian matrix of the weighted graph G is the n × n matrix L =( i,j )whoseith diagonal entry equals the sum of the weightsoftheedgesincidenttovertexi,andwhose(i, j)th off-diagonal entry equals 0 if there is no edge joining vertices i and j and equals the negative of the weight of the edge joining vertices i and j otherwise. Suppose now that G is a weighted tree on n vertices and recall that any two vertices i and j are joined by a unique path Pi,j . We define the inverse weighted distance from vertex i to vertex j as the sum