Hop Hub-Integrity of Graphs

# Sultan Senan Mahde^{1} and Abdu-Alkafi Saead Sand^{1}

^{1}Department of Mathematics, Faculty of Education and Science, University of Al-Baydha, Yemen.**Abstract:** The concept of hop hub-integrity is introduced as a new
measure of the stability of a graph $G$ and it is defined as $ H_hI(G) = min \{|S| + m(G - S)\},$
where $S$ is hop hub set and $ m(G - S)$ is the order of a maximum component of $(G - S)$. In this paper, the hop hub-integrity of some graphs is obtained.The relations between hop hub-integrity and other parameters are determined.

**Keywords:** Hub number, Hop hub number, Domination number, Connected hub number, Connected domination number.

**Cite this article as:** S. S. Mahde and A. S. Sand, *Hop Hub-Integrity of Graphs*, Int. J. Math. And Appl., vol. 9, no. 4, 2021, pp. 91-100.

**References**

- K. S. Bagga, L. W. Beineke, Wayne Goddard, M. J. Lipman and R. E. Pippert, A survey of integrity, Discrete Applied Math., (37/38)(1992), 13-28.
- K. S. Bagga, L. W. Beineke, M. J. Lipman and R. E. Pippert, Edge- integrity: A survey, Discrete math., 124(1994), 3-12.
- C. A. Barefoot, R. Entringer and H. Swart, Vulnerability in graphs - A comparative survey, J. Combin. Math. Combin. Comput., 1(1987), 12-22.
- C. A. Barefoot, R. Entringer and H. Swart, Integrity of trees and powers of cycles, Congressus Numerantium, 58(1987), 103-114.
- M. Cozzens, D. Moazzami and S. Stueckle, The tenacity of a graph, Proc. Seventh International Conference on the Theory and Applications of Graphs, New York, USA, (1995), 1111-1122.
- W. Goddard, On the vulnerability of graphs, Ph. D. Thesis. University of Natal, Durban, (1989).
- T. Grauman, S. Hartke, A. Jobson, B. Kinnersley, D. west, L. wiglesworth, P. Worah and H. Wu, The hub number of a graph, Information Processing Letters, 108(2008), 226-228.
- F. Harary, Graph theory, Addison Wesley, Massachusetts, (1969).
- T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of domination in graphs, Marcel Dckker, New York, (1998).
- S. I. Khalaf, V. Mathad and S. S. Mahde, Hubtic number in graphs, Opuscula Mathematica, 6(38)(2018), 841-847.
- S. I. Khalaf, V. Mathad and S. S. Mahde, Edge hubtic number in graphs, International Journal of Mathematical Combinatorics, 3(2018), 141-146.
- S. I. Khalaf, V. Mathad and S. S. Mahde, Hub and global hub numbers of a graph, Proceedings of the Jangjeon Mathematical Society, 23(2020), 231-239.
- V. R. Kulli, College graph theory, Vishwa International Publications, Gulbarga, India, (2012).
- S. S. Mahde and V. Mathad, A. M. Sahal, Hub-integrity of graphs, Bulletin of International Mathematical Virtual Institute, 5(2015), 57-64.
- S. S. Mahde and V. Mathad, Some operations in hub-integrity of graphs, Asia Pacific Journal of Mathematics, 2(2015), 108-123.
- S. S. Mahde and V. Mathad, Hub-integrity of splitting graph and duplication of graph element, TWMS J. App. Eng. Math., 6(2016), 289-297.
- S. S. Mahde and V. Mathad, On the weak hub-integrity of graphs, Gulf Journal of Mathematics, 5(2)(2017), 71-86.
- S. S. Mahde and V. Mathad, Hub-integrity of line graphs, Electronic Journal of Mathematical Analysis and Applications, 7(1)(2019), 140-150.
- S. S. Mahde and V. Mathad, Hub-integrity graph of graphs, Al-Baydha University Journal for Research (BUJR), 2(2020).
- A. S. Sand and S. S. Mahde, Hop hub number in graphs , submitted.
- S. K. Vaidya and L. Bijukumar, Some new families of mean graphs, Journal of Mathematics Research, 2(3)(2010), 169-176.
- S. K. Vaidya and N. J. Kothari, Some new results on domination integrity of graphs, Open Journal of Discrete Mathematics, 2(3)(2012), 96-98.
- S. K. Vaidya and N. Kothari, Domination integrity of splitting graph of path and cycle, Hindawi Publishing Corporation, ISRN Combinatorics, (2013), Article ID 795427.
- M. Walsh, The hub number of graphs, International Journal of Mathematics and Computer Science, 1(2006), 117-124.