A q-analogue of the distance matrix of a tree

The distance matrix of a bidirected tree

“…The formula for the inverse of the matrix D was obtained in a subsequent article by Graham and Lovász [6] who showed that D −1 = (e − z)(e − z) extended to a weighted tree in [1]. A q-analogue of the distance matrix was considered in [2]. In this paper, we extend the result of Graham and Lovász by considering the distance matrix for a bidirected tree, denoted D = (D ij ).…”

The product distance matrix of a tree and a bivariate zeta function of a graphs

“…The exponential distance matrix of the tree T is defined to be the n × n matrix E T = (e i,j ) 1≤i,j≤n where e i,j = q di,j . In the context of investigation related to the distance matrix of a tree, the following result was obtained by Bapat, Lal and Pati in [1]. Theorem 1.1.…”

“…For a tree T , a formula for the inverse of E T has been found in [1]. Define L q , the q-analogue of T 's laplacian as The q-analogue of the laplacian has occurred in work of other authors in different contexts as we indicate below.…”

“…Note that on setting q = 1, we get L q = L, where L is the laplacian matrix of T . For trees, the following (see [1,Proposition 3.3]) is known.…”

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

“…The exponential distance matrix of T is defined to be the n × n matrix E with (i, j)-element q d(i,j) , where q is a parameter. It was shown in [3] that the determinant of E is (1 − q 2 ) n−1 , which again is a function only of the number of vertices. The result was generalized to directed trees in [4] as follows.…”