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Title: | Combinatorial properties of a rooted graph polynomial |

Author: | Eisenstat, D.; Gordon, Gary; Redlich, A. |

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. |

URI: | http://hdl.handle.net/10385/900 |

Date: | 2008 |

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Gordon-SIAMJour ... Mathematics-vol22-2008.pdf | 318.2Kb |
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