# On the roots of all-terminal reliability polynomials

## TL;DR

This paper establishes new bounds on the roots of all-terminal reliability polynomials, showing they can have larger modulus than previously known, and identifies specific graphs with such properties.

## Contribution

It provides the first nontrivial upper bound on the modulus of all-terminal reliability roots relative to graph size and finds graphs with roots exceeding previous known moduli.

## Key findings

- Established a new upper bound on root modulus based on number of vertices.
- Discovered simple graphs with reliability roots of modulus greater than 1.
- Identified graphs with higher edge connectivity having larger reliability roots.

## Abstract

Given a graph $G$ in which each edge fails independently with probability $q\in[0,1],$ the all-terminal reliability of $G$ is the probability that all vertices of $G$ can communicate with one another, that is, the probability that the operational edges span the graph. The all-terminal reliability is a polynomial in $q$ whose roots (all-terminal reliability roots) were conjectured to have modulus at most $1$ by Brown and Colbourn. Royle and Sokal proved the conjecture false, finding roots of modulus larger than $1$ by a slim margin. Here, we present the first nontrivial upper bound on the modulus of any all-terminal reliability root, in terms of the number of vertices of the graph. We also find all-terminal reliability roots of larger modulus than any previously known. Finally, we consider the all-terminal reliability roots of simple graphs; we present the smallest known simple graph with all-terminal reliability roots of modulus greater than $1,$ and we find simple graphs with all-terminal reliability roots of modulus greater than $1$ that have higher edge connectivity than any previously known examples.