Expected rank and randomness in rooted graphs

Eisenstat, David; Feder, Jennifer; Francos, Greg; Gordon, Gary; Redlich, Amanda

doi:10.1016/j.dam.2007.08.012

Cited by 3 publications

(2 citation statements)

References 6 publications

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“…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].…”

Section: Introductionmentioning

confidence: 99%

Distinguished vertices in probabilistic rooted graphs

Gordon

Jager

2009

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The expected number of vertices that remain joined to the root vertex s of a rooted graph G s when edges are prone to fail is a polynomial EV(G s ; p) in the edge probability p that depends on the location of s. We show that optimal locations for the root can vary arbitrarily as p varies from 0 to 1 by constructing a graph in which every permutation of k -specified vertices is the "optimal" ordering for some p, 0 < p < 1. We also investigate zeroes of EV(G s ; p), proving that the number of vertices of G is bounded by the size of the largest rational zero.

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Section: Introductionmentioning

confidence: 99%

Distinguished vertices in probabilistic rooted graphs

Gordon

Jager

2009

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“…In this paper, we concentrate on combinatorial properties of the rooted graph and their connection to the polynomial. In the sequel paper [15], we turn to applications, including practical questions about optimal location for the root for a given graph, randomness, and estimation.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial Properties of a Rooted Graph Polynomial

Eisenstat¹,

Gordon²,

Redlich³

2008

SIAM J. Discrete Math.

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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.

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