Article

Determinants and Longest Cycles of Graphs

SIAM Journal on Discrete Mathematics (Impact Factor: 0.58). 01/2008; 22(3):1215-1225. DOI: 10.1137/070693898
Source: DBLP

ABSTRACT We consider the Hamiltonian cycle problem on a given graph G. With such a graph we can associate a family ℱ of probability transition matrices of Markov chains whose entries represent the probabilities of traversing corresponding arcs of the graph. When the underlying graph is Hamiltonian, we show the transition probability matrix induced by a Hamiltonian cycle maximizes — over ℱ — the determinant of a matrix that is a rank-one correction of the generator matrix of a Markov chain. In the case when the graph does not possess a Hamiltonian cycle, the above maximization yields a transition matrix of a chain with a longest simple cycle (in G) comprising that chain’s unique ergodic class. These problems also have analogous eigenvalue interpretations.

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    ABSTRACT: 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.
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    • "However, we 12 could start at one or end up at one by chance. There are also problems (see [4]) that have a form of symmetry that results in the gradient being in the space of the eigenvectors corresponding to the positive eigenvalues of H. It is a consequence of variables whose values, when they are switched, give the same objective. "
    Wiley Encyclopedia of Operations Research and Management Science, 02/2011; , ISBN: 9780470400531
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    • "It was proved in [1] that if Γ is a Hamiltonian graph, the set of global maximisers for (1) is the set of P corresponding to Hamiltonian cycles in Γ, and that the objective function of (1) has optimal value |V (Γ)| = N . Furthermore, it was proved that if Γ is a non-Hamiltonian graph, then (1) has optimal value that is strictly less than N . "
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