Girth and subdominant eigenvalues for stochastic matrices
نویسنده
چکیده
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 modulus 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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