Matrices with maximum kth local exponent in the class of doubly symmetric primitive matrices

Let A be a primitive matrix of order image, and let k be an integer with image. The kth local exponent of image, is the smallest power of A for which there are k rows with no zero entry. We have recently obtained the maximum value for the kth local exponent of doubly symmetric primitive matrices of order n with image. In this paper, we use the graph theoretical method to give a complete characterization of those doubly symmetric primitive matrices whose kth local exponent actually attain the maximum value.