A simple proof of Graham and Pollak's theorem (Q2497973)

Let \(T\) be a tree with \(n\) vertices and \(D\) the \(n\times n\) distance matrix of \(T\). \textit{R. L. Graham} and \textit{H. O. Pollack} [Bell Syst. Tech. J. 50, 2495--2519 (1971; Zbl 0228.94020)] proved the following formula: \(\det(D)=-(n-1)(-2)^{n-2}\), which is independent of the structure of \(T\). Other proofs were given by R. L. Graham with co-authors. In this paper the authors give a further proof by induction over \(n\), using the Dodgson determinant evaluation rule \(\det(A) \det(A_2)= \det(A_{11})\det(A_{nn})-\det(A_{1n})\det(A_{n1})\), where \(A\) is an \(n\times n\) matrix with \(n>2\), \(A_{ij}\) is the minor of \(A\) obtained by deleting the \(i\)th row and \(j\)th column, and \(A_2\) is the minor of \(A\) obtained by deleting rows 1 and \(n\) and columns 1 and \(n\).