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Real Reliability Roots of Simple Graphs are Dense

Mohamed Omar

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.

Real Reliability Roots of Simple Graphs are Dense

Abstract

We prove that the closure of the real roots of all-terminal reliability polynomials is exactly , 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 , where ranges over connected multigraphs and ranges over complete graphs missing an edge, are dense.

Paper Structure

This paper contains 4 sections, 5 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Future Directions

Key Result

Theorem 1.1

Let $\mathcal{R}$ be the set of real roots of all-terminal reliability polynomials of simple graphs. Then $\overline{\mathcal{R}}=[-1,0] \cup \{1\}$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: An example of the substitution construction with $n=4$. On the left, the multigraph $G$ has vertices $a$ and $b$ joined by two parallel edges. In the middle, the gadget $H_4=K_4 \backslash e$ has terminals $u$ and $v$ (so $e=uv$). On the right, each edge of $G$ is replaced by a fresh copy of $H_4$, and in each copy the terminal $u$ is identified with $a$ while the terminal $v$ is identified with $b$.

Theorems & Definitions (10)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • Proposition 3.1
  • proof : Proof of Proposition \ref{['prop:yn']}
  • proof : Proof of Theorem \ref{['thm:main']}