ELA, Volume 12, pp. 25-41, January 2005, abstract.
Girth and Subdominant Eigenvalues for Stochastic
Matrices
S. Kirkland
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 is at least 27, then for every
matrix in S(g,n), the modulus of the subdominant
eigenvalue is at least (1/5)^{1/(2 lfloor n/3 rfloor)}.