On Green's matrices of trees

The inverse C = [c(i,j)] of a irreducible nonsingular symmetric tridiagonal matrix is a so-called Green's matrix. A Green's matrix is a symmetric matrix which is given by two sequences of real umbers {u(i)} and {v(i)} such that c(i,j) = u(i)v(j) for i less than or equal to j. A similar result holds for nonsymmetric matrices. An open problem on nonsingular sparse matrices is whether there exists a similar structure for their inverses as in the tridiagonal case. Here we positively answer this question for irreducible acyclic matrices, i.e., matrices whose undirected graphs are trees. We prove that the inverses of irreducible acyclic symmetric matrices are given as the Hadamard product of three matrices, a type D matrix, a flipped type D matrix, and a matrix of tree structure which is closely related to the graph of the original matrix itself. For nonsymmetric matrices we obtain a similar structure. Moreover, our results include the result for symmetric and nonsymmetric tridiagonal matrices.