Reliability Polynomials and Their Asymptotic Limits for Families of Graphs

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.