Squared distance matrix of a weighted tree

Ravindra B. Bapat

Abstract


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.


Keywords


tree, distance matrix, squared distance matrix, determinant, inverse

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DOI: http://dx.doi.org/10.5614/ejgta.2019.7.2.8

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