ScienceDirect - Discrete Applied Mathematics : Expected rank and randomness in rooted graphs

Discrete Applied Mathematics
Volume 156, Issue 5, 1 March 2008, Pages 746-756

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doi:10.1016/j.dam.2007.08.012

Expected rank and randomness in rooted graphs

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David Eisenstat^a^,, Jennifer Feder^b^,, Greg Francos^c^,, Gary Gordon^d^, and Amanda Redlich^e^,

^aDepartment of Computer Science Princeton University, 35 Olden Street, Princeton, NJ 08540, USA

^bDepartment of Biostatistics, Johns Hopkins University, Baltimore, MD 21205, USA

^cDepartment of Mathematics, 301 Thackeray Hall University of Pittsburgh, Pittsburgh, PA 15260, USA

^dDepartment of Mathematics, Lafayette College, Easton, PA 18042, USA

^eDepartment of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA

Received 24 July 2006;

revised 14 May 2007;

accepted 12 August 2007.

Available online 10 October 2007.

Abstract

For a rooted graph G, let EV_b(G;p) be the expected number of vertices reachable from the root when each edge has an independent probability p of operating successfully. We determine the expected value of EV_b(G;p) for random trees, and include a connection to unrooted trees. We also consider rooted digraphs, computing the expected value of a random orientation of a rooted graph G in terms of EV_b(G;p). We consider optimal location of the root vertex for the class of grid graphs, and we also briefly discuss a polynomial that incorporates vertex failure.