Novel neighbourhood redefined first and second Zagreb indices on carborundum structures
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Novel neighbourhood redefined first and second Zagreb indices on carborundum structures M. C. Shanmukha1,2
· N. S. Basavarajappa2,3 · A. Usha4 · K. C. Shilpa2,5
Received: 8 July 2020 / Revised: 31 August 2020 / Accepted: 4 September 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract Graph theory offers experts with a useful tool called the topological index which analyses the characteristics of chemical compounds. Motivated by the recent work of redefined Zagreb indices (ReZ G 1 , ReZ G 2 , ReZ G 3 ), novel topological indices such as Neighbourhood redefined first and second Zagreb indices (N ReZ 1 , N ReZ 2 ) are proposed and computed for a molecular graph G of two carborundum structures. The correlation coefficient of N ReZ 1 , N ReZ 2 with ten properties of octane isomers are determined, of which entropy and acentric factor showed a good correlation. Also, the lower and upper bounds of redefined Zagreb indices are determined using classic results, Pólya–Szegö and Cauchy–Schwarz inequalities for simple graphs and trees. Keywords Silicon carbon Si 2 C3 -I and Si 2 C3 -I I · redefined Zagreb indices · Neighbourhood redefined Zagreb indices · Pólya–Szegö inequality · Cauchy–Schwarz inequality
1 Introduction and terminologies From the last 40 years, graph theory has been widely used in research areas. The topological indices are the functions, plots graph to a real-number. These graphs are used in QSAR/QSPR [14,19,20] studies to predict the physico-chemical char-
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M. C. Shanmukha [email protected]
1
Department of Mathematics, Jain Institute of Technology, Davanagere 577003, India
2
Affiliated to Visvesvaraya Technological University, Belagavi, Karnataka, India
3
Department of Mathematics, Bapuji Institute of Engineering and Technology, Davanagere 577004, India
4
Department of Mathematics, Alliance College of Engineering and Design, Alliance University, Anekal-Chandapura Road, Bangalore, Karnataka, India
5
Department of Computer Science and Engineering, Bapuji Institute of Engineering and Technology, Davanagere 577004, India
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acteristics and bio-activities of chemical compounds. In this context, chemical compounds(molecules) are expressed as graphs such that atoms depict vertices and bonds between atoms depict edges of a graph [22]. Many of the familiar degree based indices were defined earlier such as Wiener [26], Randi´c [13], Zagreb [6,7], eccentric connectivity [21], atom-bond connectivity [3], fourth atom bond connectivity [4], fifth geometric arithmetic [5] and Sanskruti [8] indices etc. Since these topological indices have wide scope [16,17], most of the work is done to determine the indices for different graphs. The recent work on redefined Zagreb indices motivates this study to define new topological indices called as Neighbourhood redefined first and second Zagreb indices (N ReZ 1 and N ReZ 2 ). The indices N ReZ 1 and N ReZ 2 of carborundum structures are computed. Here the graphs considered are simple, finite and loop
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