# Determinants and Longest Cycles of Graphs.

**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.Journal of Applied Probability 12/2011; 48(2011). DOI:10.1239/jap/1324046008 · 0.69 Impact Factor -
##### Chapter: Newton‐Type Methods

Wiley Encyclopedia of Operations Research and Management Science, 02/2011; , ISBN: 9780470400531 - [Show abstract] [Hide abstract]

**ABSTRACT:**We present an unconstrained logarithmic barrier algorithm to solve the Hamiltonian cycle problem. The interior point method described here takes advantage of significant improvements in efficiency gained by the use of a special LU decomposition. Some initial results and an example are presented to illustrate the potential effectiveness of this method.01/2010; 37(3).