On the scores and degrees in hypertournaments

A k-hypertournament H = (V, A), where V is the vertex set and A is an arc set, is a complete k-hypergraph with each k-edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the edge. In a k-hypertournament, the score s_i(losing score r_i) of a vertex is the number of edges containing v_i in which v_i is not the last element(in which v_i is the last element) and the total score of a vertex v_i is t_i = s_i − r_i. For v ∈ V we denote dH+=∑a∈Hρ(v,a)$d_H^ + = \sum\limits_{a \in H} {\rho (v,a)} $ (or simply d⁺(v)) the degree of a vertex where, ρ(v, a) is k − i if v ∈ a ∈ A and v is the ith entry in a, otherwise zero. In this paper, we obtain necessary and sufficient conditions for a k-hypertournament to be degree regular. We use the inequalities of Holder and Chebyshev from mathematical analysis to study the score and degree structure of the k-hypertournaments.