Reliability Polynomials and their Asymptotic Limits for Families of Graphs

Abstract

We present exact calculations of reliability polynomials $R(G,p)$ for lattice strips $G$ of fixed widths $L_y\le 4$ 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.

56 pages, latex, 16 figures, version to appear in J. Stat. Phys

References in corpus (4)

• Exact Potts Model Partition Function for Strips of the Square Lattice
• Complex-Temperature Phase Diagrams for the q-State Potts Model on Self-Dual Families of Graphs and the Nature of the $q\to \infty$ Limit
• Zeros of Jones Polynomials for Families of Knots and Links
• Potts Model Partition Functions for Self-Dual Families of Strip Graphs

Cited by in corpus (8)

• Spanning forests and the q-state Potts model in the limit q \to 0
• The Brown-Colbourn conjecture on zeros of reliability polynomials is false
• What is the probability of connecting two points ?
• Exact Potts/Tutte Polynomials for Polygon Chain Graphs
• Tutte polynomial of pseudofractal scale-free web
• Chromatic polynomials of random graphs
• Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions
• Exact Potts/Tutte Polynomials for Hammock Chain Graphs