A general result on the spectral radii of nonnegative k-uniform tensors

In this paper, we define k-uniform tensors for k >= 2, which are more closely related to the k-uniform hypergraphs than the general tensors, and introduce the parameter r(i)((q))(A) for a tensor A, which is the generalization of the i-th slice sum r(i) (A) (also the i-th average 2-slice sum m(i)(A)). By using r(i)((q))(A) for q >= 1, we obtain a general result on the sharp upper bound for the spectral radius of a nonnegative k-uniform tensor. When k = 2, q = 1, 2, 3, this result deduces the main results for nonnegative matrices in [1,8,27]; when k >= 3, q = 1, this result deduces the main results in [5,20]. We also find that the upper bounds obtained from different q can not be compared. Furthermore, we can obtain some known or new upper bounds by applying the general result to k-uniform hypergraphs and k-uniform directed hypergraphs, respectively.