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On the Real Reliability Roots of Graphs

Jason I. Brown, Isaac McMullin

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 $[β,0]$ where $β\approx-0.5707202942$.

On the Real Reliability Roots of Graphs

Abstract

Consider a connected graph , and assume that every edge fails independently with probability . The {\em (all-terminal) reliability polynomial} is the probability in 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 where .
Paper Structure (4 sections, 3 theorems, 30 equations, 3 figures)

This paper contains 4 sections, 3 theorems, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Almost All Graphs Have a Nonreal Reliability Root
  3. Density of the Real Reliabity Roots of Graphs
  4. Open Problems

Key Result

Proposition 2.1

Let $G$ be a connected multigraph of order $n$, size $m$ and corank $d=m-n+1 \geq 2$. If $c_{1}=0$ and $c_{2}<\frac{m(n-1)}{2d}$, then the reliability of $G$ has a nonreal root.

Figures (3)

  • Figure 1: Graph $H$.
  • Figure 2: Our gadget $K_{4}-e$ with $e=\{u,v\}$
  • Figure 3: The plot of the possible reliability roots of a graph using a $K_{4}-e$ gadget replacement for $s\in[-\frac{1}{2},-0.15]$.

Theorems & Definitions (6)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof