R. B. Bapat 2 A. K. Lal3 Sukanta Pati 4 Abstract We consider a q-analogue of the distance matrix (called the q-distance matrix) of an unweighted tree and give formulae for the inverse and the determinant, which generalize the existing formulae for the distance matrix. We obtain the Smith normal form of the q-distance matrix of a tree. The relationship of the q-distance matrix with the Laplacian...

A strongly connected digraph is called a cactoid-type if each of its blocks consisting finitely many oriented cycles sharing common directed path. In this article, we find...

Given a graph $G$, the exponential distance matrix is defined entry-wise by letting $(u,v)$-entry be $q^{\text{dist}(u,v)}$, where $\text{dist}(u,v)$ between vertices $u$ and $v$ with convention that if are in different components, then $q^{\text{dist}(u,v)}=0$. In this paper, we will establish several properties of characteristic polynomial (spectrum) for matrix, give some families graphs whic...

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....