On the Determinant of K\H{o}nig-Egerv\'ary Graphs

TL;DR

This paper introduces a new determinant factorization for K ext{"o}nig-Egerváry graphs, linking graph structure to unimodularity through a partition into perfect-flower and perfect-flower-free parts.

Contribution

It provides a novel determinant decomposition within K ext{"o}nig-Egerváry graphs, connecting unimodularity to graph structure via a specific vertex partition.

Findings

Determinant of a K ext{"o}nig-Egerváry graph factors into determinants of two induced subgraphs.

The factorization applies to the permanent as well.

The approach links unimodularity to the structure of perfect-flower and perfect-flower-free parts.

Abstract

Several graph decompositions that factorize the determinant of the adjacency matrix isolate a K\H{o}nig-Egerv\'ary 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 K\H{o}nig-Egerv\'ary graphs. In this paper we advance this point of view by introducing a new determinant factorization inside the class of K\H{o}nig-Egerv\'ary graphs. More precisely, given a K\H{o}nig-Egerv\'ary 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 K\H{o}nig-Egerv\'ary theory.