Computing Banhatti Indices of Hexagonal, Honeycomb and Derived Networks

American Journal of Mathematical and Computer Modelling 018; 3(): 38-45 http://www.sciencepublishinggroup.com/j/ajmcm doi: 10.11648/j.ajmcm.018030.11 Computing Banhatti Indices of Hexagonal, Honeycomb and Derived Networks Fazal Dayan 1, *, Muhammad Javaid 1, Muhammad Zulqarnain 1, Muhammad Tariq Ali, Bilal Ahmad 1 Department of Mathematics, University of Management and Technology, Lahore, Pakistan Department of Mathematics and Statistics, the University of Lahore, Lahore, Pakistan Email address: * Corresponding author To cite this article: Fazal Dayan, Muhammad Javaid, Muhammad Zulqarnain, Muhammad Tariq Ali, Bilal Ahmad. Computing Banhatti Indices of Hexagonal, Honeycomb and Derived Networks. American Journal of Mathematical and Computer Modelling. Vol. 3, No., 018, pp. 38-45. doi: 10.11648/j.ajmcm.018030.11 Received: February 13, 018; Accepted: May 5, 018; Published: June DD, 018 Abstract: Banhatti indices of a graph were introduced by Kulli. In this paper we have computed the general K-Banhatti indices, first and second K-Banhatti indices, K hyper Banhatti indices, modified K Banhatti indices and sum connectivity Banhatti indices for hexagonal, honeycomb and honeycomb derived networks. Keywords: Banhatti Indices, K-Banhatti Indices, K-Hyper Banhatti Indices, Modified K Banhatti Indices, Hexagonal and Honeycomb Networks 1. Introduction Kulli introduced the first and second K-Banhatti indices of a graph in [1]. These are defined as,, where indicate that the vertex and edge are incident in. Kulli also defined some properties of these newly defined indices. The coindices of K-Banhatti indices were also defined in his work. Later kulli defined K hyper-banhatti indices of V-Phenylenic nanotubes and nanotorus []. Gutman et al developed relations between Banhatti and Zagreb indices and discussed the lower and upper bounds for Banhatti indices of a connected graph in terms of Zagreb indices [3]. Kulli et al also computed Banhatti indices for certain families of benzenoid systems [4]. Moreover, Kulli introduced multiplicative hyper-banhatti indices and coindices, Banhatti geometric-arithmetic index connectivity Banhatti indices for certain families of benzenoid systems [5-7]. The Banhatti indices were also introduced and computed by Kulli [8]. Graph theory has found considerable use in modeling chemical structures. Chemical graph theory is an important branch of mathematical chemistry that uses graph theory in the mathematical modeling of chemical phenomena 1-3. Chemical graph theory has an important effect in the development of mathematical chemistry and chemical sciences. Computing topological indices in mathematical chemistry is an important branch. Topological index has become a very useful tool in the prediction of physio chemical and pharmacological properties of a compound. The number of vertices and edges are the topological index molecular structure matters. The main ingredients of the molecular topological models are topological indices which are the topological characterization of molecules by means of numerical invariants. These models are instrumental in the discovery of new applications of molecules with specific chemical, pharmacological and biological properties. Applications of graph theory have led to the emergence of a number of graph-theoretical indices [9]. These indices are used by various researchers in their studies. The first use of a topological index was made by the chemist Harold Wiener in 1947 [10]. In recent years many researchers have worked on computing topological indices [11-19]. Bharati, R. et al. have calculated the Zagreb, Randić, and ABC index of silicate, honeycomb and hexagonal networks [0]. Kulli defined atom bond connectivity reverse index of a

39 Fazal Dayan et al.: Computing Banhatti Indices of Hexagonal, Honeycomb and Derived Networks molecular graph and calculated atom bond connectivity reverse index and product connectivity reverse index for oxide and honeycomb networks [1], the direct and inverse sum Banhatti indices for some graphs [], multiplicative connectivity banhatti indices of benzenoid systems and polycyclic aromatic hydrocarbons [3] and reverse zagreb and reverse hyper-zagreb indices and their polynomials of rhombus silicate networks [4]. S. Hayat et al. [5] derived some new classes of networks from honeycomb network by using some basic graph operations like stellation, bounded dual and medial of a graph and denoted these n-dimensional honeycomb derived network of first type as 1. We give analytic formulas of the general K-Banhatti indices, first and second K-Banhatti indices, K hyper Banhatti indices, modified K Banhatti indices and sum connectivity Banhatti indices of the hexagonal, honeycomb and honeycomb derived networks.. Calculating Banhatti Indices for the Hexagonal Network Triangular, square, and hexagonal are three regular plane tessellations, composed of the same kind of regular polygons. The triangular tessellation is used to define Hexagonal network and this is widely studied in [6]. A hexagonal network is shown in figure 1. There are three types of vertices based on their degree in a hexagonal network. There are only 6 vertices of degree 3, 61 vertices of degree 4 and 3 97 vertices of degree 6. The general first and second K-Banhatti indices of a graph are defined as: (1), () where!. Figure 1. Hexagonal network. Table 1. The edge degree partition of " based on degree of end vertices. # $ %,# $ '( # $ ' *+,-./ 3,4 5 1 3,6 7 6 4,4 6 63 4,6 8 1 6,6 10 9 3330 Theorem: For a hexagonal network " of dimension, the general first and second K-Banhatti indices are given by "18 9 610 13 1310 11 14 9 333016 (3) 4 "115 0 61 4 134 13 48 9 333060 (4) Proof: Let be the graph of hexagonal network " of dimension, with 6" 3 31 and 7" 9 156, where is the number of vertices on one side of the hexagon. There are five type of edges based on degree of end vertices in a hexagonal network shown in table 1. a) We compute 8 as follows: " 18 9 610 13 1310 11 14 9 333016 b) We compute 8 as follows: "9 115 0 61 4 134 13 48 9 333060 Corollary: The first and second K-Banhatti indices for a hexagonal network " are given by "4608 1646414718 (5)

American Journal of Mathematical and Computer Modelling 018; 3(): 38-45 40 Proof: Putting =1 in equation (3), we get " 64800 3635+14014 (6) =1(17)+6(3)+10( 3)+31( )+51(9 33+30)=4608 16464+14718 Putting =1 in equation (4), we get =1(35)+6(63)+88( 3)+960( )+700(9 33+30)=64800 3635+14014 Corollary: The first and second K-hyper Banhatti indices for a hexagonal network "() are given by Proof: Putting = in equation (3), we get ("())=4608 11616+6954 (7) ("())=64800 19075+1361 (8) =1740+1614+100 3600+4080( )+4608 16896+15360=4608 11616+6954 Putting = in equation (4), we get =7500+1330+691 0736+39936 7987+64800 37600+16000 =64800 19075+1361 Corollary: The first and second modified K Banhatti indices for a hexagonal network "() are given by Proof: Putting = 1 in equation (3), we get : ("())= ; < ;; =1A 1 8 +1 9 B+6A1 10 + 1 13 B+1 10 ( 3)+1( )A1 1 + 1 Putting = 1 in equation (4), we get < + =>?; : ("())= =? + +140 (10) @? 14 B+1 8 (9 33+30)= 9 8 99 8 + 361 109 =1A 1 15 + 1 0 B+6A1 1 + 1 4 B+1 ( 3)+1( )A1 3 + 1 48 B+ 1 30 (9 33+30)= 3 10 + 1 40 +140 Corollary: The first and second sum connectivity Banhatti indices for a hexagonal network "() are given by Proof: Putting = 1 D in equation (3), we get 4 ("())=4.5 6.0340309 9.106540386 (11) 4 ("())=.33790008.1770468.39083584 (1) =1A 1 8 + 1 9 B+6A 1 10 + 1 1 B+ 13 10 ( 3)+1( )A 1 1 + 1 14 B+1 (9 33+30) = 9 +A 1 10 + 1 1 + 1 14 33 B+4+ 6 10 + 1 13 1 10 36 10 4 1 4 14 +15 Putting = 1 D in equation (4), we get =4.5 6.0340309 9.106540386 =1A 1 15 + 1 0 B+6A 1 1 + 1 1 B+ 4 4 ( 3)+1( )A 1 3 + 1 48 B+ 60 (9 33+30) = 18 60 +A 1 4 + 1 3 + 1 48 66 1 B+ 60 0 + 6 1 + 6 4 36 4 4 3 4 48 + 60 60 =.33790008.1770468.39083584 (9)

41 Fazal Dayan et al.: Computing Banhatti Indices of Hexagonal, Honeycomb and Derived Networks 3. Calculating Banhatti Indices for the Honeycomb Network 1 by adding a layer of hexagons around the boundary of 81, where determines the number of hexagons between the boundary and center of 8. Honeycomb networks are used in image processing, mobile phone base stations and computer graphics. In chemistry it is used as a representation of benzenoid hydrocarbons. A honeycomb network is shown in figure. There are 6 vertices of degree and the remaining vertices are of degree 3. Table. The edge degree partition of 8 based on degree of end vertices. # $ %,# $ '( # $ ' *+,-./, 6,3 3 11 3,3 4 9 156 Figure. Honeycomb network. A honeycomb network 8() is obtained from 8( Theorem: For a honeycomb network 8 of dimension, the general first and second K-Banhatti indices are given by 81F4 115 6 9 1567 (13) 81F4 116 9 9 1561 (14) Proof: Let be the graph of honeycomb network 8 of dimension, with 68 6 and 78 9 3. There are three types of edges based on degree of end vertices in a honeycomb network shown in table. a) We compute 8 as follows: b) We compute 8 as follows: 8 64 4 115 6 9 1567 7 1F4 115 6 9 1567 "9 14 4 116 9 9 1561 1 1F4 116 9 9 1561 Corollary: The first and second K-Banhatti indices for a honeycomb network 8 are given by Proof: Putting 1 in equation 13, we get Putting 1 in equation 14, we get 819 78 (15) 816 1801 (16) 48131149 15619 78 48180149 15616 1801 Corollary: The first and second K-hyper Banhatti indices for a honeycomb network 8 are given by Proof: Putting in equation 13, we get 888 73848 (17) 859 916516 (18)

American Journal of Mathematical and Computer Modelling 018; 3(): 38-45 4 =19731989 15688 73848 Putting in equation 14, we get 1914041889 15659 916516 Corollary: The first and second modified K Banhatti indices for a honeycomb network 8 are given by : 8 < G @ =H =H : 8 = H > = (19) (0) Proof: Putting 1 in equation 13, we get Putting 1 in equation 14, we get 311A 1 5 1 6 B 7 9 156 18 7 4 35 11 35 311A 1 6 1 9 B1 6 9 156 3 5 6 3 Corollary: The first and second sum connectivity Banhatti indices for a honeycomb network 8 are given by Proof: Putting 1 D in equation 13, we get Putting 1 D in equation 14, we get 4 86.803360514 1.07339155915.43455316 (1) 4 85.1961543 0.387544770.5651196 () 1 4 11A 1 5 1 6 B 7 9 156 18 7 I 6 1 5 5 30 7 J6 61 7 6.803360514 1.07339155915.43455316 1 4 11A 1 6 1 9 B 1 9 156 18 1 A 1 30 4 6 1 B1 6 1 1 5.1961543 0.387544770.5651196 4. Calculating Banhatti Indices for the Honeycomb Derived Networks In figure 3 a honeycomb derived network 1 with dimension 3 is given, where the black colored graph is 3- dimensional honeycomb network and blue colored graph is its stellation. Now we compute the general K-Banhatti indices, first and second K-Banhatti indices, K hyper Banhatti indices, modified K Banhatti indices and sum connectivity Banhatti indices of the honeycomb derived networks. First, we compute these indices for honeycomb derived network of first type, 1 in the following theorem. Theorem: For a honeycomb derived network 1, the general first and second K-Banhatti indices are given by Figure 3. Honeycomb derived network 1 with dimension 3.

43 Fazal Dayan et al.: Computing Banhatti Indices of Hexagonal, Honeycomb and Derived Networks (1())=1 7 +1( `1)(9 +11 )+6(10 +13 )+18( 1)(14 +15 )+(7 57+130) 16 (3) (1())=1 1 +1( `1)(18 +30 )+6(1 +4 )+18( 1)(45 +54 )+(7 57+130) 60 (4) Proof: We compute the required results as follows: (i) To compute (1()), =1(7 +7 )+1( `1)(9 +11 )+6(10 +13 )+18( 1)(14 +15 )+(7 57+130)(16 +16 ) =1 7 +1( `1)(9 +11 )+6(10 +13 )+18( 1)(14 +15 )+(7 57+130) 16 (ii) To compute (1()), =6(1 +1 )+1( `1)(18 +30 )+6(1 +4 )+18( 1)(45 +54 )+(7 57+130)(60 +60 ) =1 1 +1( `1)(18 +30 )+6(1 +4 )+18( 1)(45 +54 )+(7 57+130) 60 Corollary: The first and second K-Banhatti indices for a honeycomb derived network 1() are given by Proof: Putting =1 in equation (3), we get (1())=864 94+8 (5) (1())=340 4104+1386 (6) =1 7+1( `1)(9+11)+6(10+13)+18( 1)(14+15)+(7 57+130) 16 Putting =1 in equation (4), we get =864 94+8 =1 1+1( `1)(18+30)+6(1+4)+18( 1)(45+54)+(7 57+130) 60 =340 4104+1386 Corollary: The first and second K-hyper Banhatti indices for a honeycomb derived network 1() are given by Proof: Putting = in equation (3), we get (1())=138 17586+5946 (7) (1())=194400 93544+11410 (8) =1 49+1( `1)(81+11)+6(100+169)+18( 1)(196+5)+(7 57+130) 56 Putting = in equation (4), we get =138 17586+5946 =1 144+1( `1)(18 +30 )+6(1 +4 )+18( 1)(45 +54 )+(7 57+130) 60 =194400 93544+11410 Corollary: The first and second modified K Banhatti indices for a honeycomb derived network 1() are given by : (1())=3.375 1.15350489+13.05439 (9) Proof: Putting = 1 in equation (3), we get : (1())= ;? + = G? + H = 1 7 +1( `1)A1 9 + 1 11 B+6A1 10 + 1 13 B+18( 1)A1 14 + 1 15 B+ 16 (7 57+130) =3.375 1.15350489+13.05439 (30) Putting = 1 in equation (4), we get

American Journal of Mathematical and Computer Modelling 018; 3(): 38-45 44 = 1 1 +1( `1)A1 18 + 1 30 B+6A1 1 + 1 4 B+18( 1)A1 45 + 1 54 B+ 60 (7 57+130) = 9 10 + 3 70 +1 5 Corollary: The first and second sum connectivity Banhatti indices for a honeycomb derived network 1() are given by Proof: Putting = 1 D in equation (3), we get 4 (1())=13.5 1.8611431+5.45915517 (31) 4 (1())=6.97137003.3301604+6.87786861 (3) = 1 7 +1( `1)A1 9 + 1 11 B+6A 1 10 + 1 13 B+18( 1)A 1 14 + 1 15 B+ Putting = 1 D in equation (4), we get =13.5 1.8611431+5.45915517 = 1 1 +1( `1)A 1 18 + 1 30 B+6A 1 1 + 1 4 B+18( 1)A 1 45 + 1 54 B+ =6.97137003.3301604+6.87786861 16 (7 57+130) 60 (7 57+130) 5. Conclusion In this paper hexagonal, honeycomb and honeycomb derived network 1() are studied and the analytical closed formulas are determined for the general K-Banhatti indices, first and second K-Banhatti indices, K hyper Banhatti indices, modified K Banhatti indices and sum connectivity Banhatti indices of hexagonal, honeycomb and honeycomb derived network 1(). It will help to understand the physical features, chemical reactivities and biological activities of the hexagonal and honeycomb and honeycomb derived network 1(). These results can provide a significant determination in the pharmaceutical industry. Acknowledgements The authors would like to express their sincere gratitude to the anonymous referees for their insightful comments and valuable suggestions, which led to a number of improvements in the earlier version of this manuscript. References [1] V. R. Kulli, K-Banhatti indices of graphs, Journal of Computer and Mathematical Sciences, 7(4), 13-18, 016. [] V. R. Kulli, On K Banhatti indices and K hyper-banhatti indices of V-Phenylenic nanotubes and nanotorus, Journal of Computer and Mathematical Sciences, 7(6), 30-307, 016. [3] Gutman, V. R. Kulli, B. Chaluvaraju and H. S. Baregowda, On Banhatti and Zagreb indices, Journal of the International Mathematical Virtual Institute, 7, 53-67, 017. [4] V. R. Kulli, B. Chaluvaraju and H. S. Boregowda, Connectivity Banhatti indices for certain families of benzenoid systems, Journal of Ultra Chemistry, 13(4), 81-87, 017. [5] V. R. Kulli, Multiplicative hyper-banhatti indices and coindices of graphs, International Journal of Mathematical Archive, 7(6), 60-65, 016. [6] V. R. Kulli, A new Banhatti geometric-arithmetic index, International Journal of Mathematical Archive, 8(4), 11-115, 017. [7] V. R. Kulli, B. Chaluvaraju and H. S. Boregowda, Connectivity Banhatti indices for certain families of benzenoid systems, Journal of Ultra Chemistry, 13(4), 81-87, 017. [8] V. R. Kulli Computing Banhatti Indices of Networks, International Journal of Advances in Mathematics Volume 018, Number 1, 31-40, 018. [9] Mehdi Alaeiyan, Amir Bahrami, Javad Yazdani, Schultz Index of Nanoporous, Australian Journal of Basic and Applied Sciences, 5(6), 50-5, 011. [10] Wiener, H. Structural determination of paraffin boiling points. J. Am. Chem. Soc. 1947, 69, 17 0. [11] Gutman, I. Degree-Based topological indices. Croat. Chem. Acta, 86, 315 361, 013. [1] Garcia, I.; Fall, Y.; Gomez, G. Using topological indices to predict anti-alzheimer and antiparasitic GSK-3 inhibitors by multi-target QSAR in silico screening. Molecules 15, 5408 54, 010. [13] Furtula, B.; Gutman, I.; Dehmer, M. On structure-sensitivity of degree based topological indices. Appl. Math. Comput., 19, 8973 8978, 013.

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