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Reliability polynomials and their asymptotic limits for families of graphs

By Shu-chiuan Chang and Robert Shrock


We present exact calculations of reliability polynomials R(G,p) for lattice strips G of fixed widths Ly ≤ 4 and arbitrarily great length Lx 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

Topics: reliability polynomial, Potts model, Tutte polynomial 1
Year: 2003
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