The distance matrix of a bidirected tree

We refer to [4], [8] for basic definitions and terminology in graph theory. A tree is a simple connected graph without any circuit. We consider trees in which each edge is replaced by two arcs in either direction. In this paper, such trees are called bidirected trees. We now introduce some notation. Let e,0 be the column vectors consisting of all ones and all zeros, respectively, of the appropriate order. Let J = ee be the matrix of all ones. For a tree T on n vertices, let di be the degree of the i-th vertex and let d = (d1, d2, . . . , dn) , δ = 2e − d and z = d − e. Note that δ + z = e. Let T be a tree on n vertices. The distance matrix of a tree T is a n × n matrix D with Dij = k, if the path from the vertex i to the vertex j is of length k; and Dii = 0. The Laplacian matrix, L, of a tree T is defined by L = diag(d) − A, where A is the adjacency matrix of T. The distance matrix of a tree is extensively investigated in the literature. The classical result concerns the determinant of the matrix D (see Graham and Pollak [7]), which Stat-Math Unit, Indian Statistical Institute Delhi, 7-SJSS Marg, New Delhi 110 016, India; e-mail: [email protected] Corresponding Author: A. K. Lal, Indian Institute of Technology Kanpur, Kanpur 208 016, India; e-mail: [email protected]. Department of Mathematics, Indian Institute of Technology, Guwahati, India; e-mail: sukanta−[email protected]