On the independence polynomial of an antiregular graph

TL;DR

This paper derives explicit formulas for the independence polynomial of antiregular graphs, showing they are uniquely characterized by this polynomial within threshold graphs and analyzing its mathematical properties.

Contribution

It provides the first closed-form expressions for the independence polynomial of antiregular graphs and establishes their uniqueness and log-concavity within threshold graphs.

Findings

Independence polynomial of antiregular graphs has a closed-form expression.

Each antiregular graph is uniquely identified by its independence polynomial.

The independence polynomial is log-concave with at most two real roots.

Abstract

A graph with at most two vertices of the same degree is called antiregular (Merris 2003), maximally nonregular (Zykov 1990) or quasiperfect (Behzad, Chartrand 1967). If s_{k} is the number of independent sets of cardinality k in a graph G, then I(G;x) = s_{0} + s_{1}x + ... + s_{alpha}x^{alpha} is the independence polynomial of G (Gutman, Harary 1983), where alpha = alpha(G) is the size of a maximum independent set. In this paper we derive closed formulae for the independence polynomials of antiregular graphs. In particular, we deduce that every antiregular graph A is uniquely defined by its independence polynomial I(A;x), within the family of threshold graphs. Moreover, I(A;x) is logconcave with at most two real roots, and I(A;-1) belongs to {-1,0}.