Abstract
In this paper, 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. As an illustration of these analytical results, we exploit them to develop a new heuristic algorithm to determine a non-Hamiltonicity of a given graph.
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Notes
Henceforth, for simplicity, we call Hamiltonian transition matrices just Hamiltonian matrices.
A matrix is called doubly stochastic, if its entries are non-negative, as well as, all rows and columns sum up 1.
A graph is called cubic, if degrees of all its nodes are equal to three.
A graph is called a bridge graph, if the set \(\mathcal {V}\) can be partitioned into two non-empty sets such that there is only one arc from one partition to the other one.
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Acknowledgments
This research was partially supported by ARC Grant DP0666632. The second author was visiting INRIA in Sophia Antipolis, France, while working on this paper. We are indebted to Vladimir Ejov and Giang Nguyen for many useful discussions and for supplying the original (but different) proof of the identity in Proposition 5.3.
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Avrachenkov, K., Eshragh, A. & Filar, J.A. On transition matrices of Markov chains corresponding to Hamiltonian cycles. Ann Oper Res 243, 19–35 (2016). https://doi.org/10.1007/s10479-014-1642-2
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DOI: https://doi.org/10.1007/s10479-014-1642-2