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Proof of the Hamiltonicity-Trace Conjecture for Singularly Perturbed Markov Chains

Part of: Markov processes Polynomials and matrices Graph theory

Published online by Cambridge University Press:  14 July 2016

Vladimir Ejov ,
Nelly Litvak ,
Giang T. Nguyen  and
Peter G. Taylor
Vladimir Ejov*
Affiliation:
University of South Australia
Nelly Litvak*
Affiliation:
University of Twente
Giang T. Nguyen*
Affiliation:
University of South Australia
Peter G. Taylor*
Affiliation:
University of Melbourne
*
∗ Postal address: School of Mathematics and Statistics, University of South Australia, Mawson Lakes campus, Mawson Lakes SA 5095, Australia. Email address: ejovvl@gmail.com
∗∗ Postal address: Faculty of Electrical Engineering, Mathematics and Computer Science, Department of Applied Mathematics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands. Email address: n.litvak@ewi.utwente.nl
∗∗∗ Current address: Département d'informatique, Université Libre de Bruxelles, CP 212, Boulevard du Triomphe, 2, B-1050 Bruxelles, Belgium. Email address: giang.nguyen@ulb.ac.be
∗∗∗∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia. Email address: p.taylor@ms.unimelb.edu.au
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Abstract

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We prove the conjecture formulated in Litvak and Ejov (2009), namely, that the trace of the fundamental matrix of a singularly perturbed Markov chain that corresponds to a stochastic policy feasible for a given graph is minimised at policies corresponding to Hamiltonian cycles.

Keywords

Stochastic matrixHamiltonian cycleperturbed Markov chain

MSC classification

Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 05C45: Eulerian and Hamiltonian graphs 11C20: Matrices, determinants
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 901 - 910
Copyright
Copyright © Applied Probability Trust 2011 

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