Combinatorial Properties of a Rooted Graph Polynomial

For a rooted graph $G$, let $EV(G;p)$ be the expected number of vertices reachable from the root when each edge has an independent probability $p$ of operating successfully. We examine combinatorial properties of this polynomial, proving that $G$ is $k$-edge connected if and only if $EV'(G;1)=\cdots=EV^{k-1}(G;1)=0$. We find bounds on the first and second derivatives of $EV(G;p)$; applications yield characterizations of rooted paths and cycles in terms of the polynomial. We prove reconstruction results for rooted trees and a negative result concerning reconstruction of more complicated rooted graphs. We also prove that the norm of the largest root of $EV(G;p)$ in $\mathbb{Q}[i]$ gives a sharp lower bound on the number of vertices of $G$.