In this paper, we prove: Let A be a nonnegative primitive tensor with order m and dimension n. Then its primitive degree γ(A) ≤ (n − 1) 2 + 1, and the upper bound is sharp. This confirms a conjecture of Shao [7].

In this paper, we show that the exponent set of nonnegative primitive tensors with order m(≥ 3) and dimension n is {1, 2, . . . , (n − 1) 2 + 1}, and propose some open problems for further research.

In this paper, we present a necessary and sufficient condition for a nonnegative tensor to be a primitive one, show that the exponent set of nonnegative primitive tensors with order m(≥ n) and dimension n is {k|1 ≤ k ≤ (n − 1) 2 + 1}.

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