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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
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