DISTANCE MATRICES PERTURBED BY LAPLACIANS

Let T be a tree with n vertices. To each edge of T we assign a weight which is a positive definite matrix of some fixed order, say, s. Let D-ij denote the sum of all the weights lying in the path connecting the verticesiand j of T. We now say that D-ij is the distance between i andj. Define D := [D-ij], where D-ii is thesxsnull matrix and for i not equal j, D-ij is the distance betweeniandj. Let G be an arbitrary connected weighted graph with n vertices, where each weight is a positive definite matrix of orders. If i andjare adjacent, then define L-ij colon equals - W-ij(-1), where W-ij is the weight of the edge (i, j). Define L-ii = Sigma W-n(i not equal j,j=1)ij(-1). The Laplacian of G is now thens x ns block matrix L := [L-ij]. In this paper, we first note that D-1-L is always nonsingular and then we prove that D and its perturbation (D-1-L)(-1) have many interesting properties in common.