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Squared distance matrix of a weighted tree. (English) Zbl 1467.05152

Summary: Let \(T\) be a tree with vertex set \(\{1, \dots{}, n\}\) such that each edge is assigned a nonzero weight. The squared distance matrix of \(T\), denoted by \(\Delta \), is the \(n\times n\) 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 \(\Delta \). A formula for \(\Delta^{ - 1}\) is also obtained, under certain conditions. The results generalize known formulas for the unweighted case.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A15 Determinants, permanents, traces, other special matrix functions
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References:

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