Reliability Polynomials and their Asymptotic Limits for Families of Graphs

Details Reliability Polynomials and their Asymptotic Limits for Families of Graphs Reliability Polynomials and their Asymptotic Limits for Families of Graphs

2002-08-27

arxiv.org/abs/cond-mat/0208538v2

J. Stat. Phys. 112, 1019-1077 (2003)

Physics

Condensed Matter

Statistical Mechanics

56 pages, latex, 16 figures, version to appear in J. Stat. Phys

Scientific paper

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.

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