On the Stability of Independence Polynomials

TL;DR

This paper studies the stability of independence polynomials of graphs, showing stability for graphs with small independence number and demonstrating instability for larger graphs through specific examples.

Contribution

It establishes conditions for the stability of independence polynomials and identifies graph families with stable or unstable polynomials based on graph operations.

Findings

Independence polynomials are stable for graphs with independence number ≤ 3.

Larger graphs can have independence roots arbitrarily far to the right.

Certain graph operations preserve or destroy stability.

Abstract

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size, and its roots are called {\em independence roots}. We investigate the stability of such polynomials, that is, conditions under which the roots lie in the left half-plane (all of the real roots of independence polynomial are negative and hence lie in this half-plane). We show stability for all independence polynomials of graphs with independence number at most three, but for larger independence number we show that the independence polynomials can have roots arbitrarily far to the right. We provide families of graphs whose independence polynomials are stable and ones that are not, utilizing various graph operations.