On Symmetry of Independence Polynomials

TL;DR

This paper explores the symmetry properties of independence polynomials in graphs, demonstrating how to construct graphs with symmetric independence polynomials that relate to given graphs through specific polynomial transformations.

Contribution

It introduces a method to construct graphs with symmetric independence polynomials that contain a given graph as a subgraph, extending known results about symmetry in independence polynomials.

Findings

Constructed graphs with symmetric independence polynomials for any graph G.

Established a polynomial relation linking G, its extension, and symmetry properties.

Generalized previous results about symmetry in independence polynomials.

Abstract

An independent set in a graph is a set of pairwise non-adjacent vertices, and alpha(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while mu(G) is the cardinality of a maximum matching. If s_{k} is the number of independent sets of cardinality k in G, then I(G;x)=s_{0}+s_{1}x+s_{2}x^{2}+...+s_{\alpha(G)}x^{\alpha(G)} is called the independence polynomial of G (Gutman and Harary, 1983). If $s_{j}=s_{\alpha-j}$, 0=< j =< alpha(G), then I(G;x) is called symmetric (or palindromic). It is known that the graph G*2K_{1} obtained by joining each vertex of G to two new vertices, has a symmetric independence polynomial (Stevanovic, 1998). In this paper we show that for every graph G and for each non-negative integer k =< mu(G), one can build a graph H, such that: G is a subgraph of H, I(H;x) is symmetric, and I(G*2K_{1};x)=(1+x)^{k}*I(H;x).