Independence polynomials of graphs

TL;DR

This paper investigates the properties of independence polynomials of graphs, focusing on their evaluation at -1, symmetry, and connections to algebraic invariants, providing classifications and explicit formulas for various graph families.

Contribution

It offers new characterizations of independence polynomial values at -1, classifies symmetric polynomials in specific graph classes, and derives explicit formulas for graphs with leaves attached.

Findings

Exact values of P_G(-1) for big star graphs.

Characterization of pseudo-Gorenstein* graphs.

Classification of symmetric independence polynomials in cochordal graphs.

Abstract

In this paper, we study the independence polynomial $P_G(x)$ of a finite simple graph $G$, with emphasis on the evaluation at $x=-1$, symmetry, and its connection with the $h$-polynomial of the edge ideal of $G$. For big star graphs, we determine exactly when $P_G(-1)$ is $0, 1$, or $-1$, characterize the pseudo-Gorenstein$^*$ members, and show that there is a unique big star with symmetric independence polynomial. We also study graphs obtained from a graph $H$ by attaching leaves to selected vertices. We derive an explicit formula for the resulting independence polynomial, determine the corresponding value at $-1$, and prove that if every vertex of $H$ receives at least one leaf, then the independence polynomial is symmetric if and only if each vertex receives exactly two leaves. As an application, we obtain exact criteria for the values of $P_G(-1)$ and for the pseudo-Gorenstein$^*$ members of caterpillar graphs. For cochordal graphs, we classify all symmetric independence polynomials. Finally, for connected graphs on $n$ vertices with small independence numbers, we determine the exact range of possible values of $P_G(-1)$.