Reliability Polynomials and their Asymptotic Limits for Families of Graphs

TL;DR

This paper computes reliability polynomials for lattice strips of fixed width and infinite length, introduces a reliability per vertex concept, and analyzes the zeros of these polynomials to understand their asymptotic behavior.

Contribution

It provides exact calculations of reliability polynomials for specific lattice graphs and introduces the reliability per vertex, analyzing their zeros and asymptotic limits.

Findings

Exact reliability polynomials for lattice strips of width up to 4.

Definition and calculation of reliability per vertex for infinite graphs.

Determination of the asymptotic zero distribution and nonanalytic regions.

Abstract

We present exact calculations of reliability polynomials $R(G,p)$ for lattice strips $G$ of fixed widths $L_y \le 4$ and arbitrarily great length $L_x$ with various boundary conditions. We introduce the notion of a reliability per vertex, $r(\{G\},p) = \lim_{|V| \to \infty} R(G,p)^{1/|V|}$ where $|V|$ denotes the number of vertices in $G$ and $\{G\}$ denotes the formal limit $\lim_{|V| \to \infty} G$. We calculate this exactly for various families of graphs. We also study the zeros of $R(G,p)$ in the complex $p$ plane and determine exactly the asymptotic accumulation set of these zeros ${\cal B}$, across which $r(\{G\})$ is nonanalytic.