Combinatorial Properties of a Rooted Graph Polynomial

Abstract

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) = • • • = 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 Q[i] gives a sharp lower bound on the number of vertices of G.

Distinguished vertices in probabilistic rooted graphs

“…The fact that so much freedom is allowed for vertex location in these cases indicates how difficult is the general problem of vertex location. This is explored in some detail for other families of graphs in [1,4,10,11].…”