Operations of graphs and unimodality of independence polynomials

TL;DR

This paper introduces a new graph operation called the clique cover product, derives a formula for the independence polynomial of the resulting graph, and explores its properties related to symmetry, unimodality, and zeros, with applications to existing conjectures.

Contribution

The paper defines the clique cover product, generalizes known independence polynomial results, and studies its effects on polynomial properties like unimodality and symmetry.

Findings

Derived a formula for the independence polynomial of the clique cover product.

Proved that certain properties like unimodality and symmetry are preserved under this operation.

Solved some open conjectures related to unimodality of independence polynomials.

Abstract

Given two graphs $G$ and $H$, assume that $\mathscr{C}=\{C_1,C_2,\ldots, C_q\}$ is a clique cover of $G$ and $U$ is a subset of $V(H)$. We introduce a new graph operation called the clique cover product, denoted by $G^{\mathscr{C}}\star H^U$, as follows: for each clique $C_i\in \mathscr{C}$, add a copy of the graph $H$ and join every vertex of $C_i$ to every vertex of $U$. We prove that the independence polynomial of $G^{\mathscr{C}}\star H^U$ $$I(G^{\mathscr{C}}\star H^U;x)=I^q(H;x)I(G;\frac{xI(H-U;x)}{I(H;x)}),$$ which generalizes some known results on independence polynomials of corona and rooted products of graphs obtained by Gutman and Rosenfeld, respectively. Based on this formula, we show that the clique cover product of some special graphs preserves symmetry, unimodality, log-concavity or reality of zeros of independence polynomials. As applications we derive several known facts in a unified manner and solve some unimodality conjectures and problems.