Petersen graph is uniformly most-reliable

Abstract

A celebrated problem in network optimization is the all-terminal reliability maximization. We want to communicate a fixed number n of terminals, but we have a fixed budget constraint m. The goal is to build m links such that the all-terminal reliability is maximized in the resulting graph. In such case, the result is a uniformly most-reliable graph. The discovery of these graphs is a challenging problem that launched an interplay between extremal graph theory and computational optimization. In this paper, we mathematically prove that Petersen graph is uniformly most-reliable. The paper is closed with a conjecture on the existence of other uniformly most-reliable graphs.