# Maximum Modulus of Independence Roots of Graphs and Trees

## TL;DR

This paper investigates the maximum size of independence roots' modulus in graphs and trees, establishing asymptotic bounds that relate to the number of vertices, thereby advancing understanding of the roots' behavior.

## Contribution

It provides asymptotic bounds on the maximum modulus of independence roots for graphs and trees, a novel analysis in the spectral properties of independence polynomials.

## Key findings

- Maximum modulus of independence roots in graphs grows roughly as 3^{n/3}.
- Maximum modulus of independence roots in trees grows roughly as 2^{n/2}.
- Asymptotic bounds are established for large n.

## Abstract

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size and its roots are called independence roots. We bound the maximum modulus, $\mbox{maxmod}(n)$, of an independence root over all graphs on $n$ vertices and the maximum modulus, $\mbox{maxmod}_{T}(n)$, of an independence root over all trees on $n$ vertices in terms of $n$. In particular, we show that $$\frac{\log_3(\mbox{maxmod}(n))}{n}=\frac{1}{3}+o(1)$$ and $$\frac{\log_2(\mbox{maxmod}_{T}(n))}{n}=\frac{1}{2}+o(1).$$