Proper Colourings in r-Regular Modified Zagreb Index Graph

Abstract

In this article, the new concept proper colourings in r - regular Modified Zagreb index graph has been introduced. The first and second Modified Zagreb indices are introduced. In this article, new inequalities on chromatic number related with first and second Modified Zagreb indices are being established.

Introduction

Conclusion

In this article, new inequality has been established. Further, it has been verified for first and second Modified Zagreb indices. Finally, we conclude that new inequality on chromatic number related with Modified Zagreb indices.

References

[1] Amalorpava Jerline J., Litta E., Dhanalakshmi K., Benedict Michael Raj L.,F-Index of Generalized Mycielskian Graphs, International Journal of Recent Scientific Research, Volume 10, Issue 6, June 2019, P.No.: 32366-32371. [2] Bodendick, R. and Walther, G., On number theoretical methods in graph labelings Res.Exp.Maths (2,/1995) 3-25. [3] Bloom, D.F. Hsu, On graceful directed graphs, SIAMJ, Alg. Discrete Math.,6(1985),519-536. [4] Felix J., Litta E., Benedict Micheal Raj L., Changing and Unchanging properties of Single Chromatic Transversal Domination Number of Graphs, International Journal of Mathematics Trends and Technology, Volume 52, Issue 4, Dec 2017, P.No.: 262-266. [5] Felix J., Litta E., Benedict Micheal Raj L., Single Chromatic Transversal Dominating Irredundant Number for odd cycles, Peterson graph and Mycielski graph, Infokara Research, Volume 8, Issue 10, Oct 2019, P.No.: 139-145. [6] Felix J., Litta E., Benedict Micheal Raj L., Single Chromatic Transversal Dominating Irredundant Number of graphs, Adalya Journal, Volume 10, Issue 8, Oct 2019, P.No.: 264-272. [7] Harary, F., Graph Theory, New Delhi: Narosa Publishing House, 2001. [8] Litta E., Amalorpava Jerline J., Dhanalakshmi K.,Benedict Micheal Raj L.,and Modified Zagreb Indices of Bridge Graphs, International Journal of Mathematical Archive, Volume 8, Issue 3, Mar 2017, P.No.: 86-91. [9] Litta E., Amalorpava Jerline J., Rasika R., F- Co-index of Generalized Mycielskian Graphs, International Journal of Research in Advent Technology, Volume 7, Issue 4, Apr 2019, P.No.: 243-249. [10] Litta E., Amalorpava Jerline J., Felix J., Benedict Micheal Raj L.,First and Second Modified Zagreb Indices of Product Graphs, Infokara Research,Volume 9, Issue 1, Jan 2020, P.No.: 279-293. [11] Saridha.S. and Rajaretnam, T., \"Algebraic Properties of Plus Weighted Finite State Machine\", International Journal Of Applied Engineering Research, e-ISSN:0973-9769, p-ISSN:0973-4652, Vol.13, Number 21, 2018, 14974-14982. [12] Saridha, S. and Rajaretnam, T., \"A Study On Plus Weighted Multiset Transformation Semigroups\", International Journal Of Information And Computing Science, e-ISSN:0972-1347, Vol.6, Issue I, January 2019, 84-98. [13] Saridha, S. and Rajaretnam, T., \"On Regular Properties Of Plus Weighted Multiset Finite State Automaton\", Journal Of Applied Science And Computations, e-ISSN:1076-5131, Vol.5, Issue XII, December 2018, 87-95. [14] Saridha, S., Rajaretnam, T., Plus weighted finite state automaton, in Journal Of Computer And Mathematical Sciences (JCMS 2017), Vol.8, Issue 11, ISSN 0976-5727, pp 674-690. [15] Saridha, S. and Rajaretnam, T., \"Some properties of plus weighted multiset grammars\", International Journal Of Information And Computing Science, e-ISSN:0972-1347, Vol.6, Issue 5, May 2019, 24-37 [16] Saridha, S. and Haridha Banu . S, “A New Direction Towards Plus weighted Grammar”, International Journal for Research in Applied Science and Engineering Technology(IJRASET), ISSN: 2321 – 9653, Vol. 11, Issue II, February 2023. [17] Shalini. P, Paul Dhayabaran. D, “Proper Colourings in Magic and Anti-magic Graphs”, International Journal of Engineering and Research Technology , Vol. 3, Issue. 2, pages 815-818. February 2014. [18] Shalini. P, Paul Dhayabaran. D, “Generalization of Skolem Even Graceful Digraphs for Various Graphs”, International Journal of Mathematical Archive, 5(4), 2014, pages 65-69. [19] Shalini. P, Paul Dhayabaran. D, “Skolem Graceful Signed Graphs on Directed Graphs”, Asian Journal of Current Engineering and Maths, 3:2 March-April(2014),pages33-34. [20] Shalini. P, Paul Dhayabaran. D, “Generalization of Skolem Odd Graceful Digraphs for Various Graphs”, International Journal of Scientific and Research Technology,(2015) Volume-3, Pages 1-3. [21] Shalini. P, Paul Dhayabaran. D, An Absolute Differences of Cubic and Square Difference Labeling, International Journal of Advanced Scientific and Technical Research, May-June 2015, Issue-5, Volume-3, pages 1-8. [22] Shalini. P, Paul Dhayabaran. D, A Study on Root Mean Square Labelings in Graphs, International Journal of Engineering Science and Innovative Technology, May 2015, Volume-4, Issue-3, pages 305-309. [23] Shalini. P, Paul Dhayabaran. D, Minimization of Multiplicative Graphs, International Journal of Current Research, Volume 7, Issue-08,pages 19511-19518,August 2015. [24] Shalini. P, Gowri. R, Paul Dhayabaran. D, An absolute Differences of Cubic and Square Difference Labeling For Some Families of Graphs, International Journal of Analytical and Experimental Modal Analysis, Vol.11, Issue 10, October 2019, Pages 538 - 544, Impact Factor: 6.3. ISSN No: 0886 - 9367. [25] Shalini. P, S.A.Meena , Lehmer -4 mean labeling of graphs, International journal for research in Applied Science and Engineering Technology (IJRASET) , Volume 10, Issue XII, December 2022,Page no: 1348-1351,ISSN : 2321-9653. [26] Shalini.P, Tamizharasi.S, Power-3 Heronian Odd Mean Labeling of Graphs, International Journal for Research in Applied Science and Engineering Technology (IJRASET), Volume 10 Issue XII , December 2022, Page no: 1605-1608,ISSN:2321-9653.

Copyright

Copyright © 2023 Dr. E. Litta, S. Maragatha Dharshini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.