The exponential distance matrix of block graphs

Abstract

Let G be a connected graph with vertex set {v1,v2,⋯,vn}. As a variant of distance matrix, the exponential distance matrix was proposed by Yan and Yeh, and Bapat et al. independently. Given a nonzero indeterminate q, the exponential distance matrix F=(Fij)n×n of G is defined as Fij=qDij, where Dij is the distance between vertices vi and vj in G (i.e., the (i,j)-entry of the distance matrix of G).A graph is called a block graph, if each block is a clique (of possibly varying orders). In this paper, the determinant, inverse and cofactor sum of the exponential distance matrix of block graphs are obtained. Some known results about the exponential distance matrix and q-Laplacian matrix are generalized.