<abstract xmlns="http://www.w3.org/1999/xhtml">A <italic>k</italic>-hypertournament <italic>H = (V, A)</italic>, where <italic>V</italic> is the vertex set and <italic>A</italic> is an arc set, is a complete <italic>k</italic>-hypergraph with each <italic>k</italic>-edge endowed with an orientation, that is, a linear arrangement of the vertices contained in the edge. In a <italic>k</italic>-hypertournament, the score <italic>si</italic>(losing score <italic>ri</italic>) of a vertex is the number of edges containing <italic>vi</italic> in which <italic>vi</italic> is not the last element(in which <italic>vi</italic> is the last element) and the total score of a vertex <italic>vi</italic> is <italic>ti = si − ri</italic>. For <italic>v ∈ V</italic> we denote 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/ausi-2015-0019_i1.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mi>d</mi><mi>H</mi><mo>+</mo></msubsup><mo>=</mo><munder><mo>∑</mo><mrow><mi>a</mi><mo>∈</mo><mi>H</mi></mrow></munder><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mrow></math><tex-math>$d_H^ + = \sum\limits_{a \in H} {\rho (v,a)} $</tex-math></alternatives></inline-formula>
 (or simply <italic>d+</italic>(<italic>v</italic>)) the degree of a vertex where, <italic>ρ(v, a)</italic> is <italic>k − i</italic> if <italic>v ∈ a ∈ A</italic> and <italic>v</italic> is the ith entry in <italic>a</italic>, otherwise zero. In this paper, we obtain necessary and sufficient conditions for a <italic>k</italic>-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 <italic>k</italic>-hypertournaments.</abstract>

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 si(losing score ri) of a vertex is the number of edges containing vi in which vi is not the last element(in which vi is the last element) and the total score of a vertex vi is ti = si − ri. 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.

On the scores and degrees in hypertournaments

Eötvös Loránd University, Faculty of Informatics, Budapest, Hungary

Acta Universitatis Sapientiae, Informatica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"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...

On the scores and degrees in hypertournaments

Publié en ligne: 22 janv. 2016

Pages: 200 - 215

Reçu: 04 nov. 2015

Accepté: 22 déc. 2015

DOI: https://doi.org/10.1515/ausi-2015-0019

© 2015 Shariefuddin Pirzada et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.