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Reliability polynomials and their asymptotic limits for families of graphs. (English) Zbl 1118.82301
Summary: We present exact calculations of reliability polynomials $$R(G,p)$$ for lattice strips $$G$$ of fixed widths $$L_y \leq 4$$ 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 $$\mathcal B$$, across which $$r(\{G\})$$ is nonanalytic.

##### MSC:
 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 05B35 Combinatorial aspects of matroids and geometric lattices 05C35 Extremal problems in graph theory
##### Keywords:
Reliability polynomial; Potts model; Tutte polynomial
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