A combinatorial expression for the group inverse of symmetric M-matrices

Abstract

Abstract By using combinatorial techniques, we obtain an extension of the matrix-tree theorem for general symmetric M-matrices with no restrictions, this means that we do not have to assume the diagonally dominance hypothesis. We express the group inverse of a symmetric M–matrix in terms of the weight of spanning rooted forests. In fact, we give a combinatorial expression for both the determinant of the considered matrix and the determinant of any submatrix obtained by deleting a row and a column. Moreover, the singular case is obtained as a limit case when certain parameter goes to zero. In particular, we recover some known results regarding trees. As examples that illustrate our results we give the expressions for the Group inverse of any symmetric M-matrix of order two and three. We also consider the case of the cycle C4 an example of a non-contractible situation topologically different from a tree. Finally, we obtain some relations between combinatorial numbers, such as Horadam, Fibonacci or Pell numbers and the number of spanning rooted trees on a path.