Identities for minors of the Laplacian, resistance and distance matrices of graphs with arbitrary weights

The resistance matrix of a simple connected graph G is denoted by RG or simply by R and is defined by RG=(rij), where rij is the resistance distance between the vertices i and j in G. In this paper, we consider distance matrix of a weighted tree and the resistance matrix of any weighted graph, where the weights are nonzero scalars. We obtain the identities for minors of the Laplacian, resistance and distance matrices, which are independent of the nonsingularity of resistance and distance matrices. While finding these we obtain the necessary and sufficient condition for the resistance matrix to be singular and the rank of it. Finally, we obtain the Moore–Penrose inverse of R, when it is singular.