Product distance matrix of a graph and squared distance matrix of a tree

Abstract

Let G be a strongly connected, weighted directed graph. We define a product distance η(i, j) for pairs i, j of vertices and form the corresponding product distance matrix. We obtain a formula for the determinant and the inverse of the product distance matrix. The edge orientation matrix of a directed tree is defined and a formula for its determinant and its inverse, when it exists, is obtained. A formula for the determinant of the (entry-wise) squared distance matrix of a tree is proved.

Squared distance matrices of trees with matrix weights

“…The edge orientation matrix H = (h ij ) of T is the (n − 1) × (n − 1) matrix whose rows and columns are indexed by the edges of T and the entries are defined [4] as…”

“…The distance matrix of T , denoted by D(T ), is the n × n matrix whose rows and columns are indexed by the vertices of T and the entries are defined as follows: D(T ) = (d ij ), where d ij = d(i, j). In [4], the authors introduced the notion of squared distance matrix ∆, which is defined to be the Hadamard product D • D, that is, the (i, j)-th element of ∆ is d 2 ij . For the unweighted tree T , the determinant of ∆ is obtained in [4], while the inverse and the inertia of ∆ are considered in [6].…”

“…In [4], the authors introduced the notion of squared distance matrix ∆, which is defined to be the Hadamard product D • D, that is, the (i, j)-th element of ∆ is d 2 ij . For the unweighted tree T , the determinant of ∆ is obtained in [4], while the inverse and the inertia of ∆ are considered in [6]. In [7], the author considered an extension of these results to a weighted tree whose each edge is assigned a positive scalar weight and found the determinant and inverse of ∆.…”

Squared distance matrix of a weighted tree

“…, n} and let D be the distance matrix of T. The squared distance matrix ∆ is defined to be the Hadamard product D • D, and thus has the (i, j)-element d(i, j) 2 . A formula for the determinant of ∆ was proved in [3], while the inverse and the inertia of ∆ were considered in [4].…”

Eigenvalues of the resistance-distance matrix of complete multipartite graphs

“…Let T be a tree on n vertices with no vertex of degree 2. In [4], the authors computed the determinant of the squared distance matrix ∆(T ) of T . Further, for τ = diag(τ 1 , τ 2 , • • • , τ i ), where τ i = 2 − δ i for 1 ≤ i ≤ n and τ = τ −1 , the authors in [5], define ν = τ 1 − L(T ) τ 1 and…”