On the Real Reliability Roots of Graphs

TL;DR

This paper investigates the real roots of graph reliability polynomials, proving most graphs have nonreal roots and that these roots are densely distributed on a specific interval on the real line.

Contribution

It establishes that almost all graphs have nonreal reliability roots and characterizes the density of real roots on a particular interval.

Findings

Most graphs have at least one nonreal reliability root.

Reliability roots are dense on the interval [β, 0], with β approximately -0.5707.

The interval [β, 0] contains the accumulation points of reliability roots.

Abstract

Consider a connected graph $G$, and assume that every edge fails independently with probability $q$. The {\em (all-terminal) reliability polynomial} is the probability in $q$ that the spanning connected subgraph of operational edges is connected. In this paper we focus on the real roots of reliability polynomials ({\em reliability roots}). We prove that almost every graph has a nonreal reliability root, and that the reliability polynomials of graphs have roots dense on the interval $[\beta,0]$ where $\beta\approx-0.5707202942$.