Real Reliability Roots of Simple Graphs are Dense

TL;DR

This paper proves that the set of real roots of all-terminal reliability polynomials is exactly [-1,0] union {1}, confirming a longstanding conjecture and showing density in certain graph classes.

Contribution

It resolves a conjecture by precisely characterizing the closure of real roots of reliability polynomials and demonstrates their density in specific graph constructions.

Findings

The closure of real roots is exactly [-1,0] union {1}.

Real roots of edge-substitution graphs are dense.

The result refines previous density findings for multigraphs.

Abstract

We prove that the closure of the real roots of all-terminal reliability polynomials is exactly $[-1,0] \cup \{1\}$, resolving a conjecture of Brown and McMullin and refining the corresponding density result for multigraphs due to Brown and Colbourn. The crux of the proof is demonstrating that real reliability roots of edge-substitution graphs $G[H]$, where $G$ ranges over connected multigraphs and $H$ ranges over complete graphs missing an edge, are dense.