On the Unimodality of Independence Polynomials of Very Well-Covered Graphs

TL;DR

This paper investigates the unimodality of independence polynomials in very well-covered graphs and shows that every graph can be embedded into such a graph with a unimodal independence polynomial.

Contribution

It proves that any graph can be embedded as an induced subgraph into a very well-covered graph with a unimodal independence polynomial.

Findings

Every graph is embeddable as an induced subgraph of a very well-covered graph with unimodal independence polynomial.

The paper relates the roots of independence polynomials to their unimodality.

Supports the conjecture that all very well-covered graphs have unimodal independence polynomials.

Abstract

The independence polynomial $i(G,x)$ of a graph $G$ is the generating function of the numbers of independent sets of each size. A graph of order $n$ is very well-covered if every maximal independent set has size $n/2$. Levit and Mandrescu conjectured that the independence polynomial of every very well-covered graph is unimodal (that is, the sequence of coefficients is nondecreasing, then nonincreasing). In this article we show that every graph is embeddable as an induced subgraph of a very well-covered graph whose independence polynomial is unimodal, by considering the location of the roots of such polynomials.