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 iff EV ′(G; 1) = · · · = 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 the norm of the largest root of EV (G; p) in Q[i] gives a sharp lower bound on the number of vertices of G.