Bounds on Characteristic Polynomials

TL;DR

This paper establishes sharp bounds on partial sums of coefficients of characteristic polynomials for graphs, hyperplane arrangements, and matroids, with potential geometric interpretations and broad applicability.

Contribution

It introduces new sharp two-sided bounds for partial binomial sums of characteristic polynomial coefficients across various combinatorial structures.

Findings

Bounds hold for graph chromatic polynomials and characteristic polynomials of hyperplane arrangements.

Weak bounds extend to toric arrangements and arithmetic matroids.

The bounds are sharp and have potential geometric interpretations.

Abstract

Suppose $G$ is a simple graph with $n$ vertices, $m$ edges, and rank $r$. Let $\chi_G(t)=a_0t^n-a_1t^{n-1}+\cdots +(-1)^ra_rt^{n-r}$ be the chromatic polynomial of $G$. For $q,k\in \Bbb{Z}$ and $0\le k\le q+r+1$, we obtain a sharp two-side bound for the partial binomial sum of the coefficient sequence, that is, \[ {r+q\choose k}\le \sum_{i=0}^{k}{q\choose k-i}a_{i}\le {m+q\choose k}. \] Indeed, this bound holds for the characteristic polynomial of hyperplane arrangements and matroids, and its weak version can be generalized to the characteristic polynomial of toric arrangements and arithmetic matroids. We also propose a problem on the geometric interpretation of the above bound.