The Brown-Colbourn conjecture on zeros of reliability polynomials is false

Zeros of reliability polynomials and f-vectors of matroids

For a finite multigraph G, the reliability function of G is the probability R_G(q) that if each edge of G is deleted independantly with probability q then the remaining edges of G induce a connected spanning subgraph of G; this is a polynomial function of q. In 1992, Brown and Colbourn conjectured that for any connected multigraph G, if the complex number q is such that R_G(q)=0 then |q|<=1. We verify that this conjectured property of R_G(q) holds if G is a series-parallel network. The proof is by an application of the Hermite-Biehler Theorem and development of a theory of higher-order interla…

简单图的真实可靠性根是密集的

Real Reliability Roots of Simple Graphs are Dense

研究证明所有终端可靠性多项式实根的闭包为[-1,0]∪{1}，修正了Brown和Colbourn关于多重图密度的结果。

简单图可靠性根在单位盘中的密度

Density of reliability roots of simple graphs in the unit disk

研究简单图可靠性多项式根的分布，证明其在单位盘内稠密，并确认Brown和McMullin的猜想，利用特定图族进行证明。

Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions

I show that there exist universal constants $C(r) < \infty$ such that, for all loopless graphs $G$ of maximum degree $\le r$, the zeros (real or complex) of the chromatic polynomial $P_G(q)$ lie in the disc $|q| < C(r)$. Furthermore, $C(r) \le 7.963906... r$. This result is a corollary of a more general result on the zeros of the Potts-model partition function $Z_G(q, {v_e})$ in the complex antiferromagnetic regime $|1 + v_e| \le 1$. The proof is based on a transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of $Z_G(q, {v_e})$ to a polymer gas, followed by verification of the …

On the roots of all-terminal reliability polynomials

Given a graph $G$ in which each edge fails independently with probability $q\in[0,1],$ the all-terminal reliability of $G$ is the probability that all vertices of $G$ can communicate with one another, that is, the probability that the operational edges span the graph. The all-terminal reliability is a polynomial in $q$ whose roots (all-terminal reliability roots) were conjectured to have modulus at most $1$ by Brown and Colbourn. Royle and Sokal proved the conjecture false, finding roots of modulus larger than $1$ by a slim margin. Here, we present the first nontrivial upper bound on the modul…

斯宾塞 - 布朗极性猜想的一个反例

A counterexample for the polar conjecture of Spencer-Brown

研究针对斯宾塞 - 布朗用奇偶性传递算法证明四色定理中关于非极性五边形区域操作的断言。核心方法是给出反例。主要贡献是证明该断言错误并提出相关问题。