On the largest real root of the independence polynomial of a unicyclic graph

TL;DR

This paper investigates the extremal properties of the largest real root of the independence polynomial in unicyclic graphs, extending previous work on trees and answering open questions in the field.

Contribution

It identifies the extremal unicyclic graphs with respect to the independence polynomial order, extending prior results on trees and resolving open problems.

Findings

Determined maximum and minimum unicyclic graphs of a given order for the independence polynomial.

Identified graphs that minimize and maximize the smallest modulus of independence roots.

Disproved a conjecture by Levit and Mandrescu from 2008.

Abstract

The independence polynomial of a graph $G$, denoted $I(G,x)$, is the generating polynomial for the number of independent sets of each size. The roots of $I(G,x)$ are called the \textit{independence roots} of $G$. It is known that for every graph $G$, the independence root of smallest modulus, denoted $\xi(G)$, is real. The relation $\preceq$ on the set of all graphs is defined as follows, $H\preceq G$ if and only if $I(H,x)\ge I(G,x)\text{ for all }x\in [\xi(G),0].$ We find the maximum and minimum connected unicyclic and connected well-covered unicyclic graphs of a given order with respect to $\preceq$. This extends 2013 work by Csikv\'{a}ri where the maximum and minimum trees of a given order were determined and also answers an open question posed in the same work. Corollaries of our results give the graphs that minimize and maximize $\xi(G)$ among all connected (well-covered) unicyclic graphs. We also answer more related open questions posed by Oboudi in 2018 and disprove a conjecture due to Levit and Mandrescu from 2008.