Computation of Degree Based Indices of Basava Star Windmill Graph

**MLA Style: **B. Basavanagoud, Mahammadsadiq Sayyed. "Computation of Degree Based Indices of Basava Star Windmill Graph" International Journal of Mathematics Trends and Technology 68.1 (2022):77-85.

**APA Style: **B. Basavanagoud, Mahammadsadiq Sayyed(2022). Computation of Degree Based Indices of Basava Star Windmill Graph. International Journal of Mathematics Trends and Technology, 68(1), 77-85.

**Abstract**

In this paper, we have proposed new windmill graph, that is Basava star windmill graph. The Basava star windmill graph B_{n+2}^{(m)} is the graph obtained by taking m ≥2 copies of the graph K_{1} + K_{1,n} for n ≥1 with a vertex K_{1} in common. Further-more, we have proposed the general Sombor index of graph G. Inspired by recent work on degree-based topological indices, we have obtained first and second Zagreb index, F-index, first and second hyper-Zagreb index, harmonic index, Randic' index, general Randic' index, sum connectivity index, general sum connectivity index, atom-bond connectivity index, geometric-arithmetic index, arithmetic-geometric index, Symmetric division deg index, SK indices, general SK_{α}(G) index, general SK_{1}^{α}(G) index, Sombor index and general Sombor index of Basava star windmill graph.

**Reference**

[1] B. Basavanagoud and M. Sayyed, Certain topological indices of Basava wheel windmill graph, Submitted.

[2] L. Clark and I. Gutman, The exponent in the general Randic' index, J. Math. Chem., 43 (2008) 32-44.

[3] E. Estrada, L. Torres, L. Rodriguez and I. Gutman, An atom-bond connectiveity index: Modelling the enthalphy of formation of alkanes, Indian J. Chem., 37A (1998) 849-855.

[4] M. R. Farahani, M. R. Rajesh Kanna and R. Pradeep Kumar, On the hyper-Zagreb indices of some nanostructures, Asian Academic Research J. Multidisciplinary, 3(1) (2016) 115-123.

[5] S. Fajtlowicz, On conjectures of Graffiti-II, Congr. Numer. 60 (1987) 187-197.

[6] B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53(4) (2015) 118-1190.

[7] J. A. Gallian, A dynamic survey of graph labeling, Electron J. Combin., 15 (2008).

[8] I. Gutman and N. Trinajstic', Graph theory and molecular orbitals. Total Π electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17(4) (1972) 535-538.

[9] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem., 86 (2021) 11-16.

[10] I. Gutman, Degree-based topological indices, Croat. Chem. Acta. 86 (2013) 351-361.

[11] I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Math. Comput. Chem. 50 (2004) 83-92.

[12] I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin. (1986).

[13] F. Harary, Graph Theory, Addison-Wesley, Reading. (1969).

[14] M. Khalifeh, H. Yousefi-Azari and A. R. Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math. 157(4) (2009) 804-811.

[15] V. R. Kulli, College Graph Theory, Vishwa International Publications, Gulbarga, India (2012).

[16] V. R. Kulli, The first and second Ka indices and coindices of graphs, Int. J. Math. Archive, 7(5), (2016) 71-77.

[17] V. R. Kulli, B. Chaluvaraju and H. S. Boregowda, Some degree based connectivity indices of Kulli cycle windmill graph, South Asian J. Math., 6(6) (2016) 263-268.

[18] V. R. Kulli, B. Chaluvaraju and H. S. Boregowda, Computation of connectivity indices of Kulli path windmill graph, TWMS J. Appl. Eng. Math., 6(1) (2016) 1-8.

[19] S. Nikolic', A. Kovac'evic', Milic'evic' and N. Trinajstic', The Zagreb indices 30 years after, Croat. Chem. Acta. 76(2) (2003) 113-124.

[20] M. Randic', On characterization of molecular branching, J. Am. Chem. Soc., 97 (1975) 6609-6615.

[21] V. S. Shigehalli and R. Kanabur, Computation of new degree-based topological indices of Graphene, J. Math., (2016), 6 pages.

[22] G. H. Shirdel, H. Rezapour and A. M. Sayadi, The hyper-Zagreb index of graph operations, Iranian J. Math. Chem. 4(2) (2013) 213-220.

[23] R. Todeschini and V. Consonni, New local vertex invariants and molecular descriptors based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem. 64 (2010) 359-372.

[24] R. Todeschini, D. Ballabio and V. Consonni, Novel molecular descriptors based on functions of new vertex degrees, Novel molecular structure descriptors-Theory and applications I (I. Gutman, B. Furtula, eds.) Univ. Kragujevac. (2010) 73-100.

[25] N. Trinajstic', Chemical Graph Theory, CRC Press, Boca Raton, FL, (1992).

[26] D. Vukicevic and B. Furtula, Topological index on the ratio of geometrical and arithmetical means of end vertex degrees of edges, J. Math. Chem., 46 (2009), 1369-1376.

[27] D. Vukicevic and M. Gasperov, Bond Additive Modelling 1. Adriatic Indices,, Croatica chemica acta, 83(3) (2010), 243-260.

[28] B. Zhou and N. Trinajstic', On a novel connectivity index, J. Math. Chem. 46(4) (2009) 1252-1270.

[29] B. Zhou and N. Trinajstic', On general sum-connectivity index, J. Math. Chem. 47(1) (2010) 210-218.

**Keywords : **ABC index, hyper-Zagreb index, Randic' connectivity index, SK indices and Sombor indices, sum-connectivity index, Zagreb index, Windmill graph.