A combinatorial expression for the group inverse of symmetric M-matrices
| Autor: | Angeles Carmona, Margarida Mitjana, Andrés M. Encinas |
|---|---|
| Přispěvatelé: | Universitat Politècnica de Catalunya. Departament de Matemàtiques, Universitat Politècnica de Catalunya. MAPTHE - Anàlisi matricial i Teoria Discreta del Potencial |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
schrödinger operator
Inverse 15a10 010103 numerical & computational mathematics 01 natural sciences Combinatorics effective resistances group inverse matrix theory QA1-939 0101 mathematics Mathematics spanning tree Algebra and Number Theory Spanning tree Group (mathematics) Computer Science::Information Retrieval 15 Linear and multilinear algebra matrix theory [Classificació AMS] 010102 general mathematics 15 Linear and multilinear algebra [Classificació AMS] 15a09 Matemàtiques i estadística [Àrees temàtiques de la UPC] Expression (computer science) 05c05 Matrix tree theorem symmetric m–matrices Geometry and Topology 05 Combinatorics::05C Graph theory [Classificació AMS] matrix-tree theorem |
| Zdroj: | UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) Special Matrices, Vol 9, Iss 1, Pp 275-296 (2021) |
| DOI: | 10.1515/spma-2020-0137 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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