Some Topological Indices and their Polynomials of Graphene

ORIENTAL JOURNAL OF CHEMISTRY An International Open Access, Peer Reviewed Research Journal www.orientjchem.org ISSN: 0970-020 X CODEN: OJCHEG 2019, Vol. 35, No.(5): Pg. 1514-1518 Some Topological Indices and Their Polynomials of Graphene H. L. PARASHIVAMURTHY1, M. R. RAJESH KANNA2* and R. JAGADEESH3 1 BGS Institute of Technology, Adichunchanagiri University, B. G. Nagar-571448, Nagamangala Taluk, Mandya District, India. 2 Department of Mathematics, Sri. D. DevarajaUrs Governement First Grade College, Hunsur - 571 105, India. 3 Government Science College (Autonomous), Nrupathunga Road, Bangalore -560001, Karnataka, India. *Corresponding author E-mail: [email protected] http://dx.doi.org/10.13005/ojc/350506 (Received: June 15, 2019; Accepted: September 01, 2019) ABSTRACT In this manuscript we have computed third Zagreb index, first Zagreb polynomial, second Zagreb polynomial, third Zagreb polynomial, hyper Zagreb polynomial, forgotten index, forgotten polynomial, symmetric division index and symmetric division polynomial of Graphene. These quantities are based on degrees of the vertices. Mathematics Subject Classification: 05C12, 05C90 Keywords: Zagreb indices, Zagreb polynomials, Hyper Zagreb polynomial, Forgotten index, Forgotten polynomial, Symmetric division index, Symmetric division polynomial, Graphene. INTRODUCTION Graphene is a nanomaterial. Recently rajesh kanna and his students computed some of the topological indices of graphene1,2. In this article, we have computed third zagreb index, first zagreb polynomial, second zagreb polynomial, third zagreb polynomial, hyper zagreb index, forgotten index, symmetric division index. Also we have defined hyper zagreb polynomial, forgotten polynomial and symmetric division polynomial of graphene. index, first, second and third zagreb polynomial in 2011 as follows3. Definition 1.1: For a simple connected graph G, the third zagreb index is defined as, Definition 1. 2: The first, second and third zagreb polynomials for a simple connected graph G is defined as, Third zagreb index Fath-tabar introduced the third zagreb This is an Open Access article licensed under a Creative Commons license: Attribution 4.0 International (CC- BY). Published by Oriental Scientific Publishing Company © 2018 KANNA et al., Orient. J. Chem., Vol. 35(5), 1514-1518 (2019) 1515 rings and ‘s’ is the number of benzene rings in each row. Hyper zagreb index G. H. Shirdel et al., introduced a new distance-based zagreb indices of a graph G named hyper-zagreb index4. Definition 1. 3: The hyper zagreb index is defined as, We define hyper zagreb polynomial as follows, Fig. 1. Definition 1. 4: The hyper zagreb polynomial is defined as, Forgotten index Definition 1. 5: The forgotten topological index is also a degree based topological index, denoted by F(G) for simple graph G. It was encountered in5 and defined as, Proof: 2d structure of graphene is as shown in the above Fig. 1. Assume that it contains ‘t’ rows and ‘s’ benzene rings in every row. The edge connecting the vertices of degree di and dj is denoted by mi,j. Let |mi,j| denotes the number of edges of the type mi,j. In2 we can see that |m2,2| = (t + 4), |m2,3|=(4s + 2t - 4) and |m3,3| = (3ts - 2s- t-1). Case1: If t≠1, Consider, Definition 1. 6: The forgotten polynomial for a graph G is defined as, = (t + 4)(0) + (4s + 2t - 4)(1) + (3ts - 2s - t - 1) (0) = 4s + 2t - 4 \ ZG3 (G)= 4st + 2t - 4. Symmetric division index These topological indices are quite useful for determining total surface area and heat formation of some chemical compounds. Case2: If t=1, In2 we can find |m2,2| = 6, |m2,3| = (4s-4) and |m3,3| = (s-1) as in the following Fig. 2 below. Definition 1. 7: Symmetric division index is defined as Fig. 2. Further, we define symmetric division polynomial as follows Definition 1. 8: The symmetric division polynomial is defined as Main results Theorem 2.1. The third zagreb index of Graphene, Where ‘t’ is the number of rows of benzene Theorem 2. 2: First zagreb polynomial of Graphene, Proof. Case1: If t ≠ 1, Consider KANNA et al., Orient. J. Chem., Vol. 35(5), 1514-1518 (2019) 1516 Case 2: If t=1 Case 2: If t=1, Theorem 2. 3 The second zagreb polynomial of graphene, Theorem 2. 5 Hyper zagreb polynomial of graphene, Proof: Case1: If t ≠ 1 Proof: Case1: If t ≠ 1 Case 2: If t=1, Case 2: If t=1 Theorem 2. 4 Third zagreb polynomial of graphene, Theorem 2.6 Forgotten topological index of graphene, Proof: Case1: If t ≠1 Proof: Case 1:If t=1 KANNA et al., Orient. J. Chem., Vol. 35(5), 1514-1518 (2019) 1517 Theorem 2. 8 The symmetric division index of graphene is Proof : Case 1: for t ≠ 1 Case 2: If t=1 Theorem 2. 7 Forgotten polynomial of graphene Case 2: If t=1 is Proof: case1: If t≠1 is Case 2: If t=1 Theorem 2. 9 The symmetric division polynomial of graphene Proof: Case 1: If t≠1, KANNA et al., Orient. J. Chem., Vol. 35(5), 1514-1518 (2019) 1518 CONCLUSION In this article we have computed general formula for third zagreb index, hyper zagreb polynomial, first zagreb polynomial, second zagreb polynomial third zagreb polynomial, forgotten index, forgotten polynomial, symmetric division index and symmetric division polynomial of graphene. ACKNOWLEDGMENT Case 2: For t=1, This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Conflict of interests The authors declare that there is no conflict of interests regarding the publication of this article. Authors contributions All the authors worked together for the preparation of manuscript and all of us take the full responsibility for the content of the article, however first author typed the article and all of us read and approved the final manuscript. REFERENCES 1. 2. 3. Jagadeesh R. M.; R. Rajesh Kanna and Indumathi R. Some results on topological indices of Graphene, nanomaterials and nanotechnology., 2016, 6, 1-6. G. Shridhara.; M. R Rajesh Kanna and R. S Indumathi. Computation of Topological Indices of Graphene., Journal of Nanomaterials., 2015, 8. Ali Astanesh-asl and G. H Fath-tabar, computing first and third zagreb polynomials 4. 5. of Certain product of graphs, Iranian Jounal of Mathematical Chemistry., 2011, 2-2, 73-78. G. H Shirdel, H. Rezapour and amsayadi. Hyper zagreb index of graph operations, Iranian Journal of Mathematical Chemistry., 2013, 4(2), 213-230. B. Fur tula and Gutman. I. A forgotten topological index, Journal of Mathematical Chemistry., 2015, 53, 213-220.