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Reliability Polynomials and Their Asymptotic Limits for Families of Graphs

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Abstract

We present exact calculations of reliability polynomials R(G,p) for lattice strips G of fixed widths L y ≤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|→∞ R(G,p)1/|V| where |V| denotes the number of vertices in G and {G} denotes the formal limit lim|V|→∞ 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 \(B\), across which r({G}) is nonanalytic.

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Authors and Affiliations

  1. C. N. Yang Institute for Theoretical Physics, State University of New York, Stony Brook, New York, 11794-3840

    Shu-Chiuan Chang & Robert Shrock

  2. Department of Applied Physics, Faculty of Science, Tokyo University of Science, Tokyo, 162-8601, Japan

    Shu-Chiuan Chang

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  1. Shu-Chiuan Chang
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Chang, SC., Shrock, R. Reliability Polynomials and Their Asymptotic Limits for Families of Graphs. Journal of Statistical Physics 112, 1019–1077 (2003). https://doi.org/10.1023/A:1024663508526

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