A Simple Proof of an Inequality Connecting the Alternating Number of Independent Sets and the Decycling Number

TL;DR

This paper provides a straightforward proof of an inequality relating the independence polynomial evaluated at -1 to the decycling number of a graph, connecting independent sets and cycle removal.

Contribution

It offers an elementary and accessible proof of a known inequality linking the independence polynomial and the decycling number.

Findings

The inequality |I(G;-1)| ≤ 2^{phi(G)} holds for all graphs.

The proof simplifies understanding of the relationship between independent sets and cycle structures.

The result bridges combinatorial properties of graphs with algebraic polynomial evaluations.

Abstract

If alpha=alpha(G) is the maximum size of an independent set and s_{k} equals the number of stable sets of cardinality k in graph G, then I(G;x)=s_{0}+s_{1}x+...+s_{alpha}x^{alpha} is the independence polynomial of G. In this paper we provide an elementary proof of the inequality claiming that the absolute value of I(G;-1) is not greater than 2^phi(G), for every graph G, where phi(G) is its decycling number.