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

Let A   be a primitive matrix of order nn, and let k   be an integer with 1⩽k⩽n1⩽k⩽n. The k  th local exponent of AA, 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 1⩽k⩽n1⩽k⩽n. 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.