Perron vector bounds for a tournament matrix with applications to a conjecture of Brualdi and Li

For a primitive generalized tournament matrix, we present upper and lower bounds on an entry in its perron vector in terms of the corresponding row sum of the matrix. The bounds are then used to help prove that if n is even and sufficiently large, any tournament matrix of order n which maximizes the perron value must be almost regular. Throughout, we use both analytic and combinatorial techniques.