On the Determinant of Kőnig-Egerváry Graphs
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913067122556928 |
|---|---|
| author | Pereyra, Kevin |
| author_facet | Pereyra, Kevin |
| contents | Several graph decompositions that factorize the determinant of the adjacency matrix isolate a Kőnig-Egerváry part, such as the SD--KE decomposition and the critical independence decomposition of Larson. This suggests that the study of graph unimodularity can be approached, to a large extent, through the structure of Kőnig-Egerváry graphs. In this paper we advance this point of view by introducing a new determinant factorization inside the class of Kőnig-Egerváry graphs. More precisely, given a Kőnig-Egerváry graph $G$, we consider the partition of $V(G)$ into its perfect-flower part $PF(G)$ and its perfect-flower-free part $PFF(G)$, and prove that
\[
\det(G)=\det(G[PF(G)])\det(G[PFF(G)]).
\]
We also obtain the analogous factorization for the permanent. This decomposition provides a new tool for the study of unimodularity, reducing the problem to two induced subgraphs of a very different nature: the graph $G[PF(G)]$, whose structure is closely related to Sterboul--Deming configurations with perfect matching, and the graph $G[PFF(G)]$, which is governed by the theory of critical independent sets. In this way, the paper gives a new structural framework for the study of unimodular graphs through Kőnig-Egerváry theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_25055 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Determinant of Kőnig-Egerváry Graphs Pereyra, Kevin Combinatorics Several graph decompositions that factorize the determinant of the adjacency matrix isolate a Kőnig-Egerváry part, such as the SD--KE decomposition and the critical independence decomposition of Larson. This suggests that the study of graph unimodularity can be approached, to a large extent, through the structure of Kőnig-Egerváry graphs. In this paper we advance this point of view by introducing a new determinant factorization inside the class of Kőnig-Egerváry graphs. More precisely, given a Kőnig-Egerváry graph $G$, we consider the partition of $V(G)$ into its perfect-flower part $PF(G)$ and its perfect-flower-free part $PFF(G)$, and prove that \[ \det(G)=\det(G[PF(G)])\det(G[PFF(G)]). \] We also obtain the analogous factorization for the permanent. This decomposition provides a new tool for the study of unimodularity, reducing the problem to two induced subgraphs of a very different nature: the graph $G[PF(G)]$, whose structure is closely related to Sterboul--Deming configurations with perfect matching, and the graph $G[PFF(G)]$, which is governed by the theory of critical independent sets. In this way, the paper gives a new structural framework for the study of unimodular graphs through Kőnig-Egerváry theory. |
| title | On the Determinant of Kőnig-Egerváry Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.25055 |