Hamiltonian transition matrices

  • Authors:

  • Konstantin Avrachenkov;Ali Eshragh;Jerzy A. Filar

  • Affiliations:

  • INRIA Sophia Antipolis, France;University of South Australia, Australia;University of South Australia, Australia

  • Venue:
  • Proceedings of the 5th International ICST Conference on Performance Evaluation Methodologies and Tools
  • Year:
  • 2011

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Abstract

In this pedagogical note, we present some algebraic properties of a particular class of probability transition matrices, namely, Hamiltonian transition matrices. Each matrix P in this class corresponds to a Hamiltonian cycle in a given graph G on n nodes and to an irreducible, periodic, Markov chain. We show that a number of important matrices traditionally associated with Markov chains, namely, the stationary, fundamental, deviation and the hitting time matrix all have elegant expansions in the first n − 1 powers of P, whose coefficients can be explicitly derived. We also consider the resolvent-like matrices associated with any given Hamiltonian cycle and its reverse cycle and prove an identity about the product of these matrices.