Squared distance matrix of a weighted tree

Ravindra B. Bapat (Indian Statistical Institute, New Delhi, 110016, India)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 7, No 2 (2019): Electronic Journal of Graph Theory and Applications

Let T be a tree with vertex set {1, …, n} such that each edge is assigned a nonzero weight. The squared distance matrix of T,  denoted by Δ,  is the n × n matrix with (i, j)-element d(i, j)2,  where d(i, j) is the sum of the weights of the edges on the (ij)-path. We obtain a formula for the determinant of Δ. A formula for Δ − 1 is also obtained, under certain conditions. The results generalize known formulas for the unweighted case.