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

A Carmona,A.M Encinas,M Mitjana

doi:10.1515/spma-2020-0137

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 C 4 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.

Highlights

The classical matrix–tree theorem, whose rst version was proved by G
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
The matrix-tree theorem has been extended to the case of non-singular, symmetric, α-diagonally dominant M-matrices, see [12, 16, 26, 27]

Summary

Introduction

The classical matrix–tree theorem, whose rst version was proved by G. Observe that if Fis a spanning tree on Γλ,ω with vertex n + as a root and k edges of type {n + , i}, the forest generated in Γ, F, has k connected components and each i belongs to a di erent component and it can be considered as a root of the forest, see Figure 4 With this interpretation of F, we can state some generalization of the matrix-tree Theorem. Using the matrix–tree theorems in the preceding section we can write down a combinatorial expression for the group inverse of any irreducible symmetric M-matrix To do this we consider the associated Schrödinger operator, Lq, and we get its Green function in terms of the weights of rooted spanning forests in Γ.

The group inverse for trees

The Group inverse for C