On the Zagreb Index of T
Download PDF
Authors: T. A. NAIKOO, B. A. RATHER, U. T. SAMEE AND S. PIRZADA
DOI: 10.46793/KgJMat2402.241N
Abstract:
A tournament is an orientation of a complete simple graph. The score of a vertex in a tournament is the out degree of the vertex. The Zagreb index of a tournament is defined as the sum of the squares of the scores of its vertices. In this paper, we obtain various lower and upper bounds for the Zagreb index of a tournament.
Keywords:
Tournament, score, score sequence, Zagreb index, Landau’s theorem.
References:
[1] B. Borovićanin, K. C. Das, B. Furtula and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78(1) (2017), 17–100.
[2] R. A. Brualdi and J. Shen, Landau’s inequalities for tournament scores and a short proof of a theorem on transitive sub-tournaments, J. Graph Theory 38 (2001), 244–254. https://doi.org/10.1002/jgt.10008
[3] J. R. Griggs and K. B. Reid, Landau’s theorem revisited, Australasian J. Combin. 20 (1999), 19–24. https://ajc.maths.uq.edu.au/pdf/20/ocr-ajc-v20-p19.pdf
[4] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total f-electron energy of alternant hydrocarbons, Chemical Physics Letters 17(4) (1972), 535–538. https://doi.org/10.1016/0009-2614(72)85099-1
[5] H. G. Landau, On dominance relations and the structure of animal societies, III, the conditions for a score structure, Bull. Math. Biophys 15 (1953), 143–148.
[6] J. W. Moon, Topics on Tournaments, Holt, Rinehart and Winston, New York 1968.
[7] T. A. Naikoo, On scores in tournaments, Acta Univ. Sapientiae Informatica 10(2) (2018), 257–267. http://doi.org/10.2478/ausi-2018-0013
[8] S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient BlackSwan, Hyderabad 2012.
[9] S. Pirzada U. Samee and T. A. Naikoo, Tournaments, oriented graphs and football sequences, Acta. Univ. Sapientiae Mathematica 9(1) (2017), 213–223. https://doi.org/10.1515/ausm-2017-0014
[10] S. Pirzada, Merajuddin and U. Samee, On oriented graph scores, Matematicki Vesnik 60(3) (2008), 187–191.
[11] S. Pirzada, T. A. Naikoo and N. A. Shah, Score sequences in oriented graphs, J. Appl. Math. Comput. 23(1–2) (2007), 257–268. https://doi.org/10.1007/BF02831973
[12] S. Pirzada and U. Samee, Mark sequences in digraphs, Seminare Lotharingien de Combinatoire 55 (2006), Aricle ID B55c.
[13] K. B. Reid, Tournaments, scores, kings, generalizations and special topics, Congressus Numeratium 115 (1996), 171–211.
Login:
