On the matching arrangement of a graph and properties of its characteristic polynomial

TL;DR

This paper explores a hyperplane arrangement based on simple paths in a graph, linking it to maximum matching problems and deriving formulas for the characteristic polynomial in specific graph cases.

Contribution

It introduces a novel hyperplane arrangement related to graph paths and provides formulas for the characteristic polynomial for trees and connected graphs.

Findings

Connection established between hyperplane arrangements and maximum matching

Characteristic polynomial formula derived for trees

Reduction of problem to connected initial graphs

Abstract

This paper considers a hyperplane arrangement constructed with a subset of a set of all simple paths in a graph. A connection of the constructed arrangement to the maximum matching problem is established. Moreover, the problem of finding the characteristic polynomial is reduced to the case of a connected initial graph. The formula of the characteristic polynomial was also found for the case, when the initial graph is a tree.