# All Terminal Reliability Roots of Smallest Modulus

## TL;DR

This paper investigates the roots of all-terminal reliability polynomials of graphs, proving that for graphs with at least three vertices, the cycle graph uniquely has the root with the smallest modulus.

## Contribution

It establishes that among all graphs with n vertices, the cycle graph uniquely has the reliability polynomial root of smallest modulus, characterizing this root explicitly.

## Key findings

- Roots of smallest modulus occur only for cycle graphs $C_n$.
- The root of smallest modulus is unique for $n \\geq 3$.
- The paper provides a complete characterization of these roots.

## Abstract

Given a connected graph $G$ whose vertices are perfectly reliable and whose edges each fail independently with probability $q\in[0,1],$ the \textit{(all-terminal) reliability} of $G$ is the probability that the resulting subgraph of operational edges contains a spanning tree (this probability is always a polynomial in $q$). The location of the roots of reliability polynomials has been well studied, with particular interest in finding those with the largest moduli. In this paper, we will discuss a related problem -- among all reliability polynomials of graphs on $n$ vertices, which has a root of smallest modulus? We prove that, provided $n \geq 3$, the roots of smallest moduli occur precisely for the cycle graph $C_n$, and the root is unique.