Original Research Article

Article volume = 2021 and issue = 2

Pages: 120–128

Article publication Date: November, 1, 2021

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# Degree-Based Numerical Invariants of Grasmere Geometric Graph

#### M.C. Shanmukha(a), A. Usha(b)

(a)Department of Mathematics, Jain Institute of Technology, Davanagere-577003, Karnataka, India.

(b)Department of Mathematics, Alliance School of Applied Mathematics, Alliance University, Bangalore-562106, Karnataka, India

##### Abstract:

Graph theory plays a significant role in the applications of chemistry, pharmacy, communication, maps and aeronautical fields. This work is motivated by the old English tiles design on geometric structures, namely grasmere geometric tiles which are very popular in Victorian mosaic tiles. These are French manufactured tiles that have four different colours in it with dogtooth border. In this proposed work, the degree-based topological indices viz., Sombor index, Geometric-Harmonic index, Harmonic- Geometric index, SS index, RABC index, RABC4 index, neighborhood first Zagreb index, neighborhood second Zagreb index, neighborhood hyper Zagreb index, neighborhood third NDe index, neighborhood redefined first Zagreb index, neighborhood redefined second Zagreb index, generalized reciprocal Sanskruti index, neighborhood Sombor index and neighborhood SS index values are computed for grasmere geometric graph.

##### Keywords:

Grasmere Geometric graph, topological indices.

##### References:

- [1] Adnan Aslam, Safyan Ahmad, Muhammad Ahsan Binyamin, Wei Gao, Calculating topological indices of certain OTIS interconnection networks, Open Chem, 17 (2019), 220–228. 1
- [2] K. N. Anil Kumar, N. S. Basavarajappa, M. C. Shanmukha, A. Usha, Reciprocal Atom-bond connectivity and Fourth Atom-bond connectivity indices for Polyphenylene structure of molecules, 2(2020), 1202–1209. 10.22034/ecc.2020.255902.1095. 2.6
- [3] N. Biggs, E. K. Lloyd, R. J. Wilson, Graph Theory, Oxford University Press, (1986). 1
- [4] I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, (1986). 1
- [5] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem., 86 (2021), pp. 11–16. 2.1
- [6] Hafiz Muhammad Afzal Siddiqui, Computation of Zagreb Indices and Zagreb Polynomials of Sierpinski Graphs, Hacettepe Journal of Mathematics and Statistics, 49 (2020), 754–765. 2
- [7] V. R. Kulli, Different versions of Sombor index of some chemical structures, International Journal of Engineering Sciences & Research Technology, 10 (2021), 23–32. 2.10
- [8] Maarten van Steen, Graph Theory and Complex Networks, (2010). 1
- [9] S. Mondal, Nilanjan De, Anita Pal, Topological properties of Graphene using some novel neighborhood degree-based topological indices, International Journal of Mathematics for Industry, 11 (2019), 1950006–1950014. 2.7
- [10] M. Randic, On Characterization of Molecular Branching, J. Am. Chem. Soc., 97 (1975), 6609–6615. 1
- [11] P. S. Ranjini, V. Lokesha, I. N. Cangul, On the Zagreb indices of the line graphs of the subdivision graphs, Appl. Math. Comput., 18 (2011), 699–702. 2
- [12] M. C. Shanmukha, N. S. Basavarajappa, K. N. Anilkumar, K.C. Shilpa, Generalized Reciprocal Sanskruti Index: Chemical Applicability and Bounds, Letters in Applied NanoBioScience, 9(2020), 1595-1601. 2.9
- [13] M. C. Shanmukha, N. S. Basavarajappa, A. Usha, K. C. Shilpa, Novel neighborhood redefined First and Second Zagreb indices on Carborundum Structures, Journal of Applied Mathematics and Computing, 2020 (2020), 1–14. DOI: 10.1007/s12190-020-01435-3. 2.8
- [14] M. C. Shanmukha, A. Usha, M. K. Siddiqui, K. C. Shilpa, A. Asare-Tuah, Novel Degree-Based Topological Descriptors of Carbon Nanotubes, Journal of Chemistry, 2021 (2021), 15 pages. https://doi.org/10.1155/2021/3734185. 2.3
- [15] G. H. Shirdel, H. Rezapour, A. M. Sayadi, The hyper-Zagreb index of graph operations, Iran. J. Math. Chem., 4(2013), 213–220. 1
- [16] Sourav Mondal, Nilanjan De, Anita Pal, On Some Neighborhood Degree-Based Indices for Some Oxide and Silicate Networks, J. Multidisciplinary Scientific Journal, 2 (2019), 384–409. 2
- [17] N. Trinajstic, Chemical Graph Theory, CRC Press, Boca Raton, FL. (1992). 2
- [18] R. E. Ulanowicz, Quantitative methods for ecological network analysis, Comput. Biol. Chem., 28 (2004), 321–339. 1
- [19] A. Usha, M. C. Shanmukha, K. N. Anil Kumar, K.C. Shilpa, Comparision of novel index with geometric-arithmetic and sum-connectivity indices, 11 (2021), 5344–5360. 2.2
- [20] Weidong Zhao, M. C. Shanmukha, A. Usha, Mohammad Reza Farahani, K.C. Shilpa, Computing SS Index of Certain Dendrimers, Journal of Mathematics, 2021 (2021), 14 pages. https://doi.org/10.1155/2021/7483508. 2.4
- [21] Zeeshan Saleem Mufti, Asma Wajid, Tanweer UI Islam, Nasir Ali, A Consequential Computation of Degree Based Topological Indices of Grasmere Geometric Graph, Punjab University Journal of Mathematics, 52 (2020), 45–60. 1

##### Cite this article as:

- M.C. Shanmukha, A. Usha, Degree-Based Numerical Invariants of Grasmere Geometric Graph, Communications in Combinatorics, Cryptography & Computer Science, 2021(2), PP.120–128, 2021
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