Girth and subdominant eigenvalues for stochastic matrices

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Stephen J. Kirkland

Abstract

The set S (g, n) of all stochastic matrices of order n whose directed graph has girth g is considered. For any g and n, a lower bound is provided on the modules of a subdominant eigenvalue of such a matrix in terms of g and n, and for the cases g=1, 2, 3 the minimum possible modulus of a subdominant eigenvalue for a matrix in S (g, n) is computed. A class of examples for the case g= 4 is investigated, and it is shown that if g > 2n/3 and n ≥ 27,then for every matrix in S(g, n),the modulus of the subdominant eigenvalue is at least (1/5)1/(2[n/3]).

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