On f-Symmetries of the Independence Polynomial

TL;DR

This paper investigates the f-symmetry of the independence polynomial in graphs formed by the corona operation, revealing conditions under which these polynomials are f-symmetric, symmetric, and unimodal, especially when H has specific properties.

Contribution

It establishes that the independence polynomial of the corona of a graph G with certain graphs H is f-symmetric, extending previous results on symmetry and unimodality.

Findings

I(G*H;x) is f-symmetric when H has p vertices, q edges, and independence number 2.

For H = K_{r}-e, the independence polynomial is symmetric and unimodal with a unique mode.

Generalizes earlier results on symmetry and unimodality of independence polynomials for specific graph constructions.

Abstract

An independent set in a graph is a set of pairwise non-adjacent vertices, and a(G) is the size of a maximum independent set in the graph G. 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_a*x^a,a=a(G), is called the independence polynomial of G (I. Gutman and F. Harary, 1983). If s_{a-i}=f(i)*s_{i} holds for every i, then I(G;x) is called f-symmetric (f-palindromic). If f(i)=1, then I(G;x) is symmetric (palindromic). The corona of the graphs G and H is the graph G*H obtained by joining each vertex of G to all the vertices of a copy of H. In this paper we show that if H is a graph with p vertices, q edges, and alpha(H)=2, then I(G*H;x) is f-symmetric for some elegant function f. In particular, if H = K_{r}-e, we show that I(G*H;x) is symmetric and unimodal, with a unique mode. This finding generalizes results due to (Stevanovic, 1998) and (Mandrescu, 2012) claiming that I(G*(K_2-e);x)=I(G*2K_1;x) is symmetric and unimodal for every graph G.