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arXiv · 2603.09059

On the Real Reliability Roots of Graphs

Jason I. Brown·Isaac McMullin
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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 $β\approx-0.5707202942$.

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Jason I. Brown, Isaac McMullin. 2026-03-10. On the Real Reliability Roots of Graphs. https://arxiv.org/abs/2603.09059

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