Zagreb Polynomials and Redefined Zagreb Indices of silicon-carbon Si2C

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1 Zagreb Polynomials and Redefined Zagreb Indices of silicon-carbon Si2C I and Si C II p, Young Chel Kwun, Abaid ur Rehman Virk, Waas Nazeer,, Wei Gao and Shin Min Kang 1. Department of Mathematics, Dong A Uniersity, Korea; yckwun@dau.ac.kr 2. Department of Mathematics, Uniersity of Management and Technology, Lahore Pakistan; abaid.math@gmail.com. Department of Mathematics, Diision of Science and Technology, Uniersity of Education Lahore Pakistan 4. School of Information Science and Technology, Yunnan Normal Uniersity, Kunming , China; gaowei@ynnu.edu.cn 5. Department of Mathematics and RINS, Gyeongsang National Uniersity, Jinju 52828, Korea, smkang@gnu.ac.kr Corresponding Author: nazeer.waas@ue.edu.pk ; Abstract The application of graph theory in chemical and molecular structure research far eceeds people's epectations, and it has recently grown eponentially. In the molecular graph, atoms are represented by ertices and bonded by edges. In this report, we study the silicon-carbon Si2C I and Si 2C II. We compute seeral Zagreb polynomials and Redefined Zagreb indices of these silicon-carbons. Keywords: Silicon, Zagreb inde, Zagreb polynomial. 1. Introduction Silicon has many adantages oer other semiconductor materials: it is minimal effort, it is non-toic, basically its accessibility is unlimited, and it is inoled in purging, deelopment for many years behind it. It is used for all cutting-edge electronics badget The most reliable structures of two-dimensional (2D) silicon-carbon monolayer mies with different stoichiometric blends were epected in [1] which in light of the molecule swarm streamlining signified as (PSO) method joined with thickness utilitarian hypothesis optimization. Graphene sheets are effectiely limited to [2,], and from this point on, this honeycomb structure's 2D material is ery much due to its surprising mechanical, electronic and optical properties (including its anomalous uantum effect). To a large etent, it aroused a strong interest in research lobby, oerwhelming electronic conductiity and 2018 by the author(s). Distributed under a Creatie Commons CC BY license.

2 high mechanical uality. In particular, the intriguing electronic properties of graphene make this 2D material potentially useful for faster, smaller electronic products. Like carbon, silicon also has a honeycomb structure of two-dimensional allotropes, particularly siliconene. So far, a series of efforts hae been made to open a band gap on the sheets of silicene. In addition, a two-dimensional silicon-carbon (Si-C) monolayer can be iewed as an undoped two-dimensional carbon monolayer - graphene and an undoped two-dimensional monolayer of silicon - between the siliconene Tuning material. A great deal of effort has been read [4,5] to predict the most stable structure of SiC sheet for more data. In this paper we study Zagreb polynomials and Redefined Zagreb indices of Silicon Carbide structures. 2. Basic Definitions Many studies hae shown that there is a strong intrinsic link between the chemical properties of chemical compounds and drugs (such as boiling point and melting point) and their molecular structure. The topological inde defined on the structure of these chemical molecules can help researchers better understand the physical characteristics, chemical reactiity and biological actiity. Therefore, the study of topological indices of chemical substances and chemical structures of drugs can make up for the lack of chemical eperiments and proide theoretical basis for the preparation of drugs and chemical substances. In the past two decades, a large number of digital map inariants (topological indices) hae been defined and used for correlation analysis in theoretical chemistry, pharmacology, toicology and enironmental chemistry. The first and second Zagreb indices are one of the oldest and most well-known topological indices defined by Gutman in 1972 and are gien different names in the literature, such as the Zagreb group inde, Sag. Loeb group parameters and the most common Zagreb inde. The Zagreb inde is one of the first indices introduced and has been used to study molecular compleity, chirality, ZE isomers and heterogeneous systems. The Zagreb inde shows the potential applicability of deriing multiple linear regression models. The first and the second Zagreb indices [6] are defined as

3 M ( G) ( d d ) 1 ue ( G) u and M ( G) ( d d ). 2 ue ( G) u For details see [7]. Considering the Zagreb indices, Fath-Tabar ([8]) defined first and the second Zagreb polynomials as du d M1( G, ) ue ( G) and du d M 2( G, ). ue ( G) The properties of M 1 ( G, ), M 2 ( G, ) polynomials for some chemical structures hae been studied in the literature [9,10]. After that, in [11], the authors defined the third Zagreb inde M ( G) ( d d ). ue ( G) u and the polynomial du d M ( G, ). ue ( G) In the year 2016, [12] following Zagreb type polynomials were defined ue ( G) d ( d d ) u u M 4( G, ) ue ( G) d ( d d ) u M 5( G, ) adu bd M a, b ( G, ) ue ( G)

4 ' du ad b M a, b( G, ). ue ( G) Ranjini et al. [1] redefines the Zagreb inde, ie, the redefined first, second and third Zagreb indices of graph G. These indicators appear as u Re ZG1 G ue PD1 d u d d d du d ReZG 2 G d d ue PD u G d d d d ReZG. ue PD 1 1 u u For details about topological indices and its applications we refer [15-22].. Silicon Carbide Si C I p, Figure 1 and Figure 2 show two-dimensional molecular diagrams of silicon carbide Si C I. In order to describe its molecular graph, we hae set this way: we define p as the number of connected units in a row (chain), and as the number of connected rows, the number of p cells per connection. In Figures and 4, we demonstrate how cells are connected in one row (chain) and how one row is connected to another row. We will use Si C I p to represent this molecular graph., Figure 1. Unit Cell of Si C I p,

5 Figure 2. Sheet of Si C I p for p=4 and =, Figure. Sheet of Si C I p for p=4 and =1, A Figure 4. Sheet of Si C I p for p=4 and =2, In figures 1-4 Carbon atom C are shown as brown and Silicon atom Si are shown as blue.

6 Theorem 1. Let Si C I p be the Silicon Carbide. Then, M 1 S i2 C I, p p p 9 p M 2 Si2C I, 2 p 2 4 6p p 9p M Si C I p p p 2, = M 4 Si2C I p p p p 5. M Si C I p p p p a b, = Proof , , M Si C I p p p p, a 2a 2b a 15 p p 5 1. a, b a b p p b M ' Si C I, p 2 6p 8 9 1a 2b 1a b 2a 2b 2a b We can see from [14] that the total number of ertices are 10 p and total number of edges are 15p 2 p. The edge set of Si C I p with 1 has following fie partitions,, And Si2 C I p Si2 C I p Si2C I p 2, Si2C I E 1,2 e u E, d u 1,? d, E 1, e u E, d u 1,? d, E 2,2 e u E, d u 2,? d, 爘 E e u E d u 2 牋, d., u 爘 E e E Si C I d u,? d. Now E 1,2 1, E 1, 1, E2,2 p 2, E 2, 6p 1 8 1,

7 And E, 15 p 9p M1( Si2C,, ) u I p ue Si C I ue{1,2} Si2C I ue 1, Si2C I ue 2,2 Si2C I 2 ue Si2C I ue Si2C I 2, d d , 2, 2, 5 6 2, 2,, 4 4 1,2 1, 2,2 E Si C I p E Si C I p E Si C I p E Si C I p E Si C I p 2,, = p 2 6 p 1 8( 1) 15p 9p p 2 1 6p p 9p M ( Si C I, ) u 2 u ESi2 C I u E{1,2} Si2 C I u E 1, Si2C I u E 2,2 Si2C I u E d d 2, Si2C I u E, Si2C I ,,, 6 9 2, 2, 4 1,2 1, 2,2 E Si C I p E Si C I p E Si C I p E Si C I p E Si C I p 2,, p2 6p8 9 15p 9p1 7.

8 . M( Si2C,, ) u I p ue Si C I ue {1,2} Si2C I ue 1, Si2C I ue 2,2 Si2C I ue 2, d d Si2 C I ue, Si2C I 2, 2, 2, 1 0 2, 2, ,2 1, 2,2 E Si C I p E Si C I p E Si C I p E Si C I p E Si C I p 2,, 2 = p 2 6p p 9p p p p = 15 p 8p 11 7 p ( 2,, ) d u d u M Si C d I p ue Si C I ue {1,2} Si2C I ue 1, Si2C I ue 2,2 Si2C I ue , Si2C I ue, Si2C I 2, 2, 2, , 2, 4 8 1,2 1, 2,2 E Si C I p E Si C I p E Si C I p E Si C I p E Si C I p 2,, p 2 6p p 9p

9 d d d 5. M5 Si2C I, u u E Si C I u E{1,2} Si2C I u E 1, Si2C I u E 2,2 Si2C I 2, ue Si2C I u E Si2C I 2, 2, 2, , 2,, ,2 1, 2,2 E Si C I p E Si C I p E Si C I p E Si C I p E Si C I p 2,, p2 6p8 9 15p 9p1 7. ad bd 6. M, 2,, u a b Si C I p u E Si C I a2b ab 2a2b u E{1,2} Si2 C I u E 1, Si2C I u E 2,2 Si2 C I E 2, 2, u E 2, Si C I p u E, Si C I p 2, 2, 2, 2a b a b Si2 C I E Si2 C I 18 2a b a b a2b ab 2a2b 1,2 1, 2,2 E Si C I p E Si C I p E Si C I p 2, , a2b ab 2a2b 2a b a b p p p p a 2a 2b =2 7p10 9 p2 1 15p 9p p p5 1 b a

10 d 7. (, ) ua d b M Si C I ` ab, u E Si2 C I 1 a2b 1 a b 2a 2b u E u E{1,2} Si 2 C I u E {1,} Si2 C I u E Si 2C I 2a b 2, Si2 C I u E, Si2 C I a b 2,2 1 a 2b 1 a b 2a 2b 1,2 2 1, 2 2,2 2 E Si C I E Si C I E Si C I 2a b a b 2, 2, E Si C I E Si C I 1 a 2 b 1 a b 2 a 2 b 2 a b a b p2 6p8 9 15p 9p1 7. Theorem 2. Let Si C I p be the Silicon Carbide. Then,, 1. Re ZG Si C I 15 p Re ZG2 Si2C I 15 p p Proof: Re ZG Si C I 15 p 290 p

11 Re ZG Si C I ue Si2C I du d d d u , 2, 2, ue 1,2 Si C I p ue 1, Si C I p ue 2,2 Si C I p 2, 2, ue 2, Si C I p u E, Si C I p E Si2C 1,2 I E Si2C 1, I E 2,2 Si2C I E 2, Si2C I E Si2C, I ( p 2 ) 6p p 9p p 156 p p 6 15 p.

12 Re ZG Si C I ue Si2C I du d d d u , 2, 2, ue 1,2 Si C I p ue 1, Si C I p ue 2,2 Si C I p 2, 2, ue 2, Si C I p u E, Si C I p E Si2C 1,2 I E Si2C 1, I E 2,2 Si2C I E 2, Si2C I E Si2C, I ( p 2 ) 6p p 9p p 1 p p p p p u u. Re ZG Si C I d d d d u E Si2 C I 2 u E1,2 Si2 C I u E 1, Si2C, I p u E 2,2 Si2 C I u E 2, Si2 C I u E, Si2 C I Si C I 6 E Si C I 12 E Si C I 16 E Si C I 1,2 1, 2,2 0 E Si C I 54 E 2,, p p p p p 290 p Silicon Carbide Si2C-I I[ ] Figure 5-8 shows a two-dimensional molecular map of silicon carbide Si2C II. In order to describe its molecular graph, we set this way [14]: we define p as the number of unit

We will use Si C II p to represent this molecular graph., Figure 5. One Unit of Si C II p, Figure 6.

13 cells connected in a row (chain), and with, we represent the number of connected rows, the number of rows per connection It is p units. In Figures 7 and 8, we demonstrate how cells are connected in a row (chain) and how a row is connected to another row. We will use Si C II p to represent this molecular graph., Figure 5. One Unit of Si C II p, Figure 6. Sheet of Si C II p for p= and =, Figure 7. Sheet of Si C II p for p=5 and =1, Figure 8. Sheet of Si2C II for p=5 and =2 Theorem 4 Let Si C II p be the Silicon Carbide. Then,

14 M 1 Si2C II, 2 2 p p p 1 p M 2 Si2C II, p 4 2 4p p 1p M Si C II p p p 2, M 4 Si2C II p p p p , M Si C II p p p p , , = M Si C II p p p p a, b a 2a 2b a b 15 p 5 p 5. 1a 2b 1a b 2a 2b 2a b M ' Si C II, =2 2 p 2 4p 4 7 a, b a b p p Proof: :Let G be the graph of Si C II p where we define p as the number of connected, unit cells in a row (chain) and by we represents the number of connected rows each with p number of cells.from the graph of Si C II p we can see that the total number, of ertices are 10 p and total number of edges are 15p p. The edge set of Si C II p with 1 has following fie partitions,, And Si2 C II p Si2 C II p Si2C II p 2, Si2C I p E 1,2 e u E, d u 1,? d, E 1, e u E, d u 1,? d, E 2,2 e u E, d u 2,? d, 爘 E e u E I, d u 2?? d., Si2C I p 爘 E e u E I, d u,? d. Now E 1,2 2, E 1, 1, E2,2 2p 2, E2, 8p 8 14,

15 And E, 15 p 1p M1( Si2C,, ) u II p ue Si C II ue {1,2} Si2C II ue 1, Si2C II ue 2,2 Si2C II ue 2, d d Si2C II ue, Si2C II 2, 2, 2, 5 6 2, 2, 4 4 1,2 1, 2,2 E Si C II p E Si C II p E Si C II p E Si C II p E Si C II p 2,, =2 2p 2 8p p 1p p p p 1p M2( Si2 C,, ) u II p u E Si C II u E{1,2} Si2C II u E 1, Si2C II u E 2,2 Si2C II u E 2, d d Si2C II u E, Si2C II 2,,, 6 9 2, 2, 4 1,2 1, 2,2 E Si C II p E Si C II p E Si C II p 2,, E Si C II p E Si C II p p2 8p p 1p p 2 4p4 7 15p 1p 1 11.

16 . M( Si2C,, ) u II p ue Si C II d d ue {1,2} Si2C II ue 1, Si2C II ue 2,2 Si2C II 2 ue 2, Si2C II ue, Si2C II 2, 2, 2, 1 0 2, 2, ,2 1, 2,2 E Si C II p E Si C II p E Si C II p E Si C II p E Si C II p 2,, 2 =2 2 p 2 8p p 1p p p p = 15p 11p p d 4. 4(,, ) u du d M Si C II p ue Si C II ue {1,2} Si2C II ue 1, Si2C II ue 2,2 Si2C II ue 2, Si2C II ue, Si2C II 2, 2, 2, , 2, 4 8 1,2 1, 2,2 E Si C II p E Si C II p E Si C II p E Si C II p E Si C II p 2,, p 2 8p p 1p p 2 4p p 1p

17 d d d 5. M5 Si2C II, u u E Si C II u E{1,2} Si2C II u E 1, Si2C II u E 2,2 Si2C II u E 2, Si2C II u E, Si2C II 2, 2, 2, , 2, ,2 1, 2,2 E Si C II p E Si C II p E Si C II p E Si C II p E Si C II p 2,, p2 8p p 1p = 2 2 p 2 4p p 1p ad bd 6. M, 2,, u a b Si C II p u E Si C II a2b ab 2a2b u E{1,2} Si2 C II u E 1, Si2 C II u E 2,2 Si2 C II 2, 2, u E 2, Si C II p u E, Si C II p 2,, 2, 2a b a b 2, 2, 2a b a b a2b ab 2a2b 1,2 1, 2,2 E Si C II p E Si C II p E Si C II p E Si C II p E Si C II p 2,, a2b ab 2a2 b 2a b a b 2 2p2 8p p 1p 1 11 a = 10p p 5p5 2a 2b a b p p p a 2a 2b a = 2 5p5 7 2 p1 15p 1 p1 11 p p b

18 d 7. (, ) ua d b M Si C II ` ab, u E Si2 C II 1 a2b 1 a b 2a 2b u E 2a b Si2 C II u E, Si2 C II u E{1,2} Si2 C II u E{1,} Si2 C II u E Si 2C II 2, a b 2,2 1 a 2b 1 a b 2a 2b 1,2 1, 2,2 E Si C II E Si C II E Si C II 2a b a b 2, 2, E Si C II E Si C II 1 a2b 1 a b 2a 2b 2a b a b 2 2p2 8p p 1p a2b 1 a b 2a 2b 2a b a =2 2 p 2 4p4 7 15p 1 p1 11 b. Theorem 2: Let Si C II p be the Silicon Carbide. Then,, Re ZG Si C II 15 p Re ZG2 Si2C II 15 p p Proof: Re ZG Si C II 15 p 40 p

19 Re ZG Si C II ue Si2C II du d d d u , 2, 2, ue 1,2 Si C II p ue 1, Si C II p ue 2,2 Si C II p 2, 2, ue 2, Si C II p u E, Si C II p E Si2C 1,2 II E 1, Si2C II E 2,2 Si2C II E 2, Si2C II E, Si2C II ( ) (2p 2 ) 8p p 1 p p 2 p p p 15 p.

20 Re ZG Si C II ue Si2C II du d d d u , 2, 2, ue 1,2 Si C II p ue 1, Si C II p ue 2,2 Si C II p 2, 2, ue 2, Si C II p u E, Si C II p E Si2C 1,2 II E 1, Si2C II E 2,2 Si2C II E 2, Si2C II E, Si2C II ( ) (2p 2 ) 8p p 1 p p 2 p p p p p u u. Re ZG Si C II d d d d u E Si2 C II u E 1,2 Si2 C II u E 1, Si2C, II p u E 2,2 Si2 C II u E 2, Si2C II u E, Si2 C II Si2C II 54 E Si2 C II 6 E Si C II 12 E Si C II 16 E Si C III 0 E 1,2 1, 2,2 2,, p p p p 2(6) p 40 p

21 Concluding Remarks In this paper, we hae computed seeral algebraic Zagreb type polynomials and first, second and third Zagreb indices for the silicon-carbon Si2C I and Si 2C II. Our results are important for the researcher working in this area of research and applicable in many areas of applied sciences, for eample, chemistry, physics, electronics, etc. References 1. Li, P.; Zhou, R.; Zeng, X.C. The search for the most stable structures of silicon carbon monolayer compounds. Nanoscale 2014, 6, Nooselo, K.S.; Geim, A.K.; Morozo, S.V.; Jiang, D.; Zhang, D.Y.; Dubonos, S.V.; Grigoriea, I.V.; Firso, A.A. Electric field effect in atomically thin carbon films. Science 2004, 06, Nooselo, K.S.; Mccann, E.; Morozo, S.V.; Falko, V.I.; Katsnelson, M.I.; Zeitler, U.; Jiang, D.; Schedin, F.; Geim, A.K. Unconentional uantum Hall effect and Berry s phase of 2pi in bilayer graphene. Nat. Phys. 2006, 2, Li, Y.; Li, F.; Zhou, Z.; Chen, Z. Highly Stable Chiral (A)6 B Supramolecular Copolymer. A Multialency-Based Self-Assembly Process. J. Am. Chem. Soc. 2011, 1, Zhou, L.; Zhang, Y.; Wu, L. SiC2 Siligraphene and Nanotube. Noel Donor Materials in Ecitonic Solar Cells. Nano Lett. 201, 1, Das, K. C., Xu, K., & Nam, J. (2015). Zagreb indices of graphs. Frontiers of Mathematics in China, 10(), I. Gutman, K. C. Das, The first Zagreb inde 0 years after, MATCH-Commun. Math. Comput. Chem. 50 (2004) G. H. Fath-Tabar, Old and new Zagreb inde, MATCH-Commun. Math. Comput. Chem. 65 (2011) Gutman I, Das KC. The first Zagreb inde 0 years after. MATCH Commun Math Comput Chem 2004; 50: Gutman I. New bounds on zagreb indices and the zagreb co-indices. Bol Soc Paran Mat 201; 1(1): G. H. Fath-Tabar, Zagreb Polynomial and Pi Indices of some Nano Structures, Digest Journal of Nanomaterials and Biostructures 4-1 (2009)

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PAijpam.eu ON TOPOLOGICAL INDICES FOR THE LINE GRAPH OF FIRECRACKER GRAPH Ashaq Ali 1, Hifza Iqbal 2, Waqas Nazeer 3, Shin Min Kang 4

PAijpam.eu ON TOPOLOGICAL INDICES FOR THE LINE GRAPH OF FIRECRACKER GRAPH Ashaq Ali 1, Hifza Iqbal 2, Waqas Nazeer 3, Shin Min Kang 4 International Journal of Pure and Applied Mathematics Volume 6 No. 4 07, 035-04 ISSN: 3-8080 printed ersion); ISSN: 34-3395 on-line ersion) url: http://www.ijpam.eu doi: 0.73/ijpam.6i4.8 PAijpam.eu ON