Density of reliability roots of simple graphs in the unit disk

Abstract

Brown and Colbourn (1992) showed that the complex roots of the reliability polynomial of connected multigraphs are dense in the unit disk and that the closure of the real roots is $[-1,0] \cup \{1\}$. We prove the simple graph analogues of both results, confirming a recent conjecture of Brown and McMullin. The proof uses the family of graphs $C_m[K_n]$ obtained by substituting each edge of a cycle $C_m$ with a complete graph $K_n$, and relies on the asymptotic behavior of the reliability and split reliability polynomials of $K_n$.