A note on Frobenius inner product and the \(m\)-distance matrices of a tree

Abstract

The \(m\)-distance matrix \(D_m\) of a simple connected undirected graph has an important role in computing the distance matrix \(D\) of the graph from the powers of the adjacency matrix using Hadamard product. This paper shows that for an undirected tree \(T\) with diameter \(d,\left\{D_0 . D_1, \ldots, D_d\right\}\) is an orthogonal basis for the space spanned by the binary equivalent matrices of the first \(d+1\) powers of the adjacency matrix of \(T\) and it gives an invertible conversion matrix for finding the \(m\)-distance matrix of \(T\) using Frobenius inner product on matrices.