### 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.

Original language | English |
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Pages (from-to) | 1019-1077 |

Number of pages | 59 |

Journal | Journal of Statistical Physics |

Volume | 112 |

Issue number | 5-6 |

DOIs | |

Publication status | Published - 2003 Sep 1 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Chang, S. C., & Shrock, R. (2003). Reliability Polynomials and Their Asymptotic Limits for Families of Graphs.

*Journal of Statistical Physics*,*112*(5-6), 1019-1077. https://doi.org/10.1023/A:1024663508526