Abstract. 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 4 2 n+ vertices and consists of one (4 1) n + -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.

A two-colored directed digraph D is primitive if and only if there exist nonnegative integers h and k with h+k>0 such that for each pair (i, j) of vertices there is a (h, k)-walk in D from i to j. A (h, k)-walk from i to j consisting of h red arcs and k blue arcs. The exponent of the primitive two-colored digraph D, denoted exp(D), is defined to be the smallest value of h+k over all such h and k. A class of two-colored digraphs with two cycles whose uncolored digraph has n vertices and consists of one n-cycle and one 3-cycle is considered. The upper bound of primitive exponent and characteristic of extremal two-colored digraphs are given.

So far, many problems of the primitive exponent of traditional single nonnegative matrix have been resolved. Currently, pushing traditional single nonnegative matrix to matrix pairs which have relatively application in many areas such as information science, communication networks, computer science, is a new trend. According to correspondence relation, two-colored digraphs can be used to solve matrix problems. We consider the uncolored digraph of two-colored digraph has vertices and none non-common arc. The range of exponents and extremal two-colored digraphs are given.

Abstract. There is a one-to-one relationship between nonnegative matrix pairs and two-colored digraph. With the knowledge of graph theory, by studying the associated directed digraph of a class of special nonnegative matrix pairs, that is a class of two-colored digraphs whose uncolored digraph have n vertices and consists of one n − cycle and one m − cycle are considered. The upper bound of exponent and characteristic of extreme two-colored digraphs are given.