Determinants and Longest Cycles of Graphs

ArticleinSIAM Journal on Discrete Mathematics 22(3):1215-1225 · January 2008with15 Reads
Impact Factor: 0.65 · 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.