It is a brand-new research in combinatorial matrix theory to extend the exponent of traditional single nonnegative primitive matrix to the exponent of nonnegative primitive matrix pairs. With the knowledge of graph theory, the problem of primitive exponent of nonnegative matrix pairs can be transformed into the associated directed digraph of nonnegative matrix pairs, that is two-colored digraphs. A class of two-colored digraphs whose uncolored digraph has 4n+2 vertices and consists of one (4n + 1)-cycle and one n-cycle is considered. The primitive conditions, the upper bound, the lower bound, and the characterizations of extremal two-colored digraphs are given. The results provide a basis for the study of the exponent of nonnegative primitive matrix pairs and the exponent of nonnegative primitive matrix tuples in the general case.
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