Article Some Algebraic Polynomials and Topological Indices of Octagonal Network

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1 Article Some Algebraic Polynomials and Topological Indices of Octagonal Network Young Chel Kwun, Waqas Nazeer 2, Mobeen Munir 2 and Shin Min Kang 3,4, * Department of Mathematics, Dong-A Uniersity, Busan , Korea; yckwun@dau.ac.kr 2 Diision of Science and Technology, Uniersity of Education, Lahore 54000, Pakistan; nazeer.waqas@ue.edu.pk (W.N.); mmunir@ue.edu.pk (M.M.) 3 Department of Mathematics and RINS, Gyeongsang National Uniersity, Jinju 52828, Korea 4 Center for General Education, China Medical Uniersity, Taichung 40402, Taiwan * Corresponding author: smkang@gnu.ac.kr Abstract: M-polynomial of different molecular structures helps to calculate many topological indices. A topological index of graph G is a numerical parameter related to G which characterizes its molecular topology and is usually graph inariant. In the field of quantitatie structure-actiity (QSAR) quantitatie structure-actiity structure-property (QSPR) research, theoretical properties of the chemical compounds and their molecular topological indices such as the Zagreb indices, Randic index, Symmetric diision index, Harmonic index, Inerse sum index, Augmented Zagreb index, multiple Zagreb indices etc. are correlated. In this report, we compute closed forms of M-polynomial, first Zagreb polynomial and second Zagreb polynomial of Octagonal network. From the M-polynomial we recoer some degree-based topological indices for Octagonal network. Moreoer, we gie a graphical representation of our results. Keywords: M-polynomial; Zagreb polynomial; topological index; network. Introduction Chemical reaction network theory is an area of applied mathematics that attempts to model the behaior of real-world chemical systems. Since its foundation in the 960s, it has attracted a growing research community, mainly due to its applications in biochemistry and theoretical chemistry. It has also attracted interest from pure mathematicians due to the problems that arise from the mathematical structures. Cheminformatics is an emerging field in which quantitatie structure-actiity (QSAR) and Structure-property (QSPR) relationships predict the biological actiities and properties of nanomaterial see [-4]. In these studies, some physcio-chemical properties and topological indices are used to predict bioactiity of the chemical compounds see [5-7]. The branch of chemistry which deals with the chemical structures with the help of mathematical tools is called mathematical chemistry. Chemical graph theory is that branch of mathematical chemistry which applies graph theory to mathematical modeling of chemical phenomena. In chemical graph theory a molecular graph is a simple graph (haing no loops and multiple edges) in which atoms and chemical bonds between them are represented by ertices and edges respectiely. A graph GV (, E ) with ertex set V(G) and edge set E(G) is connected, if there exist a connection between any pair of ertices in G.

2 2 of 8 A network is simply a connected graph haing no multiple edges and loops. A chemical graph is a graph whose ertices denote atoms and edges denote bonds between those atoms of any underlying chemical structure. The degree of a ertex is the number of ertices which are connected to that fixed ertex by the edges. In a chemical graph the degree of any ertex is at most 4. The distance between two ertices u and is denoted as du (, ) = d ( u, ) and is the length of shortest path between u and in graph G. The number of ertices of G, adjacent to a gien ertex, is the degree of this ertex, and will be denoted by d. The concept of degree in graph theory is closely related (but not identical) to the concept of alence in chemistry. For details on basics of graph theory, any standard text such as [3] can be of great help. G Seeral algebraic polynomials hae useful applications in chemistry such as Hosoya Polynomial (also called Wiener polynomial) [8] that plays a ital role in determining distance-based topological indices. Among other algebraic polynomials, M-polynomial [4], introduced in 205 plays the same role in determining many degree-based topological indices. Definition. Let G be a simple connected graph. The M-polynomial of G is defined as: (,, ) m ( ) i j M G x y ij G x y δ i j Δ =. Where δ = Min{ d V (G)}, Δ= Max{ d V (G)}, and m ( G ) is the edge u E( G) such that { d, d } = { i, j}. u ij This polynomial is one of the key areas of interest in computational aspects of materials. From this M-polynomial, we can calculate many topological indices. M-polynomial of different molecular structures has been computed in [9-2]. The topological index of a molecule structure can be considered as a non-empirical numerical quantity which quantifies the molecular structure and its branching pattern in many ways. In this point of iew, the topological index can be regarded as a score function which maps each molecular structure to a real number and is used as a descriptor of the molecule under testing [3-6]. Topological indices gie good predictions of the ariety of physico-chemical properties of chemical compounds containing boiling point, the heat of eaporation, heat of formation, chromatographic retention times, surface tension, apor pressure etc. Since the 970s, two degree based graph inariants hae been extensiely studied. These are the first Zagreb index M and the second Zagreb index M2, introduced by Gutman and Trinajstic [7] 2 and are defined as: M ( G) = (d ) and M ( G) V( G) 2 = du d. u E ( G) Results obtained in the theory of Zagreb indices are summarized in the reiew [8]. Second modified Zagreb index is defined as:

3 3 of 8 m M 2 ( G) = u E ( G). d d In 998, working independently, Bollobas and Erdos [9] and Amic et al. [20] proposed general Randic index. It has been extensiely studied by both mathematicians and theoretical chemists (See, for example, [2,22]). The Randic index is defined as: where α is an arbitrary real number. Symmetric diision index is defined as: ( ) SDD G u R ( G) ( dud) α α =, u E( G) = u E ( G) min( du, d) max( du, d) +. max( du, d) min( du, d) Another ariant of Randic index is the harmonic index defined as: The Inerse sum index is defined as: The augmented Zagreb index is defined as: 2 HG ( ) =. d + d u E ( G) dud IG ( ) =. d + d u E ( G) dud AG ( ) = du + d 2 u E ( G) and it is useful for computing heat of formation of alkanes [23,24] These topological indices can be recoered from M-polynomial [4], see following table. u u 3 Table. Deriation of some degree-based topological indices from M-polynomial Topological Index Deriation from M(G; x, y) First Zagreb (D +D ) M(G; x, y) Second Zagreb (D D ) M(G; x, y) Second Modified Zagreb (S S ) M(G; x, y) General Randic α N (D D ) M(G; x, y) Symmetric Diision Index (D S +S D ) M(G; x, y) Harmonic Index 2 S J (M(G ; x,y)) x x=

4 4 of 8 Inerse sum Index S J D D(M(G ; x,y)) x x y x= Augmented Zagreb Index S Q J D D (M(G ; x, y)) x 2 x y x= where x y ( ) ( ) ( ) ( ) ( f x, y ( f x, y f t, y f x, t Dx = x,? Dy = y? Sx = dt Sy = dt?? J( f ( x y) ) = f ( x, x),? x y t t 0 0 α ( ( )) = ( ) Qα f xy, x f xy,. In 203, Shirdel et al. in [25] proposed hyper-zagreb index which is also degree based index. Definition 2.Let G be a simple connected graph. Then the hyper-zagreb index of G is defined as ( ) HM G [ ] 2 = du+ d. u E( G) In 202 Ghorbani and Azimi [26] proposed two new ariants of Zagreb indices. Definition 3. Let G be a simple connected graph. Then the first multiple Zagreb index of G is defined as ( G) [ ] PM = du+ d. u E ( G) We refer [27] to the readers for more detail. Definition 4.Let G be a simple connected graph. Then the second multiple Zagreb index of G is defined as ( G) [ ] PM 2 = du+ d. u E ( G) Definition 5. Let G be a simple connected graph. Then the first Zagreb polynomial of G is defined as ( ) [ d d, u+ M G x = ] x u E ( G). Definition 6. Let G be a simple connected graph. Then second Zagreb polynomial of G is defined as ( ) [ d d, u+ M G x = ] x u E ( G) 2. In this article, we compute M-polynomial, first Zagreb polynomial and second Zagreb polynomial of Octagonal network shown in figure. We also compute some degree-based topological indices.

5 5 of 8 We denote octagonal network by O{ nm, } for nm, 2. The planer representation of O{ nm, } is shown in figure with m rows and n columns of octagonal. Let V is the ertex set and E is the edge set of O { nm, }. Then j 3j 2 V = { xi ; i 2n, i is odd and j 3m+ } { xi ; i 2 n, i is een and j m+} 3j 3j { x2n, x2n; j m} and j j+ 3j 2 3j 2 i i i i+ 3j 2 3j 3j 3j+ i i+ i i j j+ { x2nx2 n ; j 3 m} E = { x x ; i 2n, i is odd and j 3 m} { x x ; i 2n, i is odd and j m+} { x x ; i 2n 2, i is een and j m} { x x ; 3 i 2n, i is odd and j m} The number of ertices in octagonal network is ( ) octagonal network is ( ) 6m+ n+ m. MAIN RESULTS In this part we gie our main computational results. 4m+ 2 n+ 2m and number of edges in an Theorem. Let O{ nm, } be the octagonal network. Then the M-polynomial of O { nm, } is ( ) = M DPZ x y n m x y n m x y mn n m x y n,, (2 2 4) (4 4 8) ( ).

6 6 of 8 Figure. The Octagonal network O { nm, } Proof. Let O { nm, } be the octagonal network. The edge set EO ( {, nm} ) is diided into three edge partitions based on degrees of end ertices. The first edge partition E ( O { nm, }) contains 2n+ 2m+ 4 edges u, where d = d = 2. The second edge partition E2 ( O { nm, }) contains 4n+ 4m 8 edges u, where d = 2, d = 3. The third edge partition 3 { nm, } u u E ( O ) contains 6mn 5n 5m+ 4 edges u, where d = d = 3. From definition u the M-polynomial of O { nm, } is gien by

7 7 of 8 (?, ) M O x y = m x y { nm, } i j ij i j m22x y m23x y m33x y m22x y m23x y + m33x y u E ( O{ nm, }) u E2 ( O{ nm, }) u E3 ( O{ nm, }) = + + = ( { nm, }) 2( { nm, }) 3( { nm, }) = E O x y + E O x y + E O x y = (2n+ 2m+ 4) x y + (4n+ 4m 8) x y + (6mn 5n 5m+ 4) x y. Figure 2. 3D and implicit plot of the M-polynomial of O {4,5} Now we compute some degree-based topological indices from this M-polynomial. Proposition 2. Let O {, nm} be the octagonal network. Then. M ( O{ nm, }) = 36mn-2m-2 n. 2. M2 ( O { nm, }) = 54mn-3m-3n M ( ) = n+ m+ + mn m 3. 2 O{, nm} 4. ( nm ) a a a R α O{, } = (2n+ 2m+ 4)4 + (4n+ 4m 8)6 + (6mn 5m 5n+ 4 )9. 5. Rα ( O{ nm, }) (2n+ 2m+ 4) (4n+ 4m 8) (6mn 5m 5n+ 4) = + +. α α α 4 6 9

8 8 of SSD( ) = n + m - + 2mn O{, nm} O = n+ m+ + 2 mn H( {, nm} ) I( O ) = - n- m+ + 9 mn {, nm} A( O ) = - n- m+ + mn {, nm} Proof. Let M( O{, }; x, y) = f( x, y ) = (2n+ 2m+ 4) x y + (4n+ 4m 8) x y + (6mn 5n 5m+ 4) x y. nm Then Dx f( x, y) = 2 (2n+ 2m+ 4) x y + 2(4n+ 4m 8) x y + 3(6mn 5n 5m+ 4) x y, Dy f( x, y) = 2 (2n+ 2m+ 4) x y + 3(4n+ 4m 8) x y + 3(6mn 5n 5m+ 4) x y, DDf y x ( xy, ) = 4 (2n+ 2m+ 4) x y + 6(4n+ 4m 8) x y + 9(6mn 5n 5m+ 4) xy, (2n+ 2m+ 4) 2 2 (4n+ 4m 8) 2 3 (6mn 5n 5m+ 4) 3 3 Sy ( f( x, y)) = x y + x y + x y, (2n+ 2m+ 4) 2 2 (4n+ 4m 8) 2 3 (6mn 5n 5m+ 4) 3 3 SS x y( f( x, y) ) = x y + x y + x y, y α ( ( x, y) ) = 2 α ( ) + 3 α ( ) + 3 α (6 5 5m+ 4) x y D f n m x y n m x y mn n α α 2α 2 2 α α 2 3 2α 3 3 Dx Dy ( f( x, y)) = 2 (2n+ 2m+ 4) x y (4n+ 4m 8) x y + 3 (6mn 5n 5m+ 4) x y α (2n+ 2m+ 4) 2 2 (4n+ 4m 8) 2 3 (6mn 5n 5m+ 4) 3 3 Sy ( f( x, y) ) = x y + x y + x y, α α α (2n 2m 4) 2 2 (4n 4m 8) 2 3 (6mn 5n 5m 4) 3 3 Sx α Sy α ( f( x, y)) = + + x y + + x y + + x y, 2α α α 2α (4n+ 4m 8) D ( f( x, y) ) = (2n+ 2m+ 4) x y + x y + ( 6mn 5n 5m+ 4) x y, 3 S y x 2 2 3(4n+ 4m 8) D ( f( x, y) ) = (2n+ 2m+ 4) x y + x y + ( 6mn 5n 5m+ 4) x y,. 2 S x y Jf ( xy, ) = (2n+ 2m+ 4) x + (4n+ 4m 8) x + (6mn 5n 5m+ 4) x, (2n+ 2m+ 4) 4 (4n+ 4m 8) 5 (6mn 5n 5m+ 4) 6 Sx Jf( x, y) = x + x + x, JDD x y f( xy, ) = 4(2n+ 2m+ 4) x + 6(4n+ 4m 8) x + 9(6mn 5n 5m+ 4) x, 4 6(4n+ 4m 8) 5 9(6mn 5n 5m+ 4) 6 SxJDxDy f( x, y ) = (2n+ 2m+ 4) x + x + x, 5 6

9 9 of Dy f( x, y) = 2 (2n+ 2m+ 4) x y + 3 (4n+ 4m 8) x y + 3 (6mn 5n 5 m+ 4) x y, Dx Dy f( x, y ) = 2 (2n+ 2m+ 4) x y ( 4n+ 4m 8) x y + 3 (6mn 5n 5m+ 4) x y, JDx Dy f( x, y) = 2 (2n+ 2m+ 4) x (4n+ 4m 8) x + 3 (6mn 5n 5 m+ 4) x, Q 2 JDx Dy f( xy, ) = 2 (2n+ 2m+ 4) x (4n+ 4m 8) x + 3 ( 6mn 5n 5 m+ 4) x, (6mn 5n 5m + 4) Sx JDx Dy f( x, y) = 2 (2n+ 2m+ 4) x + 2 ( 4n+ 4 m 8) x + x 4, 3 4. { nm, } ( x y) M ( O ) = D + D f( x, y) = 36mn-2m-2 n. x= y= Figure 2. 3D and implicit plot for the first Zagreb index 2. M2 ( O{ nm, }) = Dy Dx( f ( x, y)) = 54mn-3m-3n+ 4. x= y= Figure 3. 3D and implicit plot for the second Zagreb index

10 0 of 8 3. m 2 M 2 ( O{ nm, }) = SxSy( f( x, y)) = n+ m+ + mn. x= y= Figure 4. 3D and implicit plot for the modified second Zagreb index 4. ( {, nm} ) α α Rα O = D D ( f( x, y)) = (2n+ 2m+ 4)4 + (4n+ 4m 8)6 + ( 6mn 5m 5n+ 4)9. x y x= y= a a a 5. α α ( O{ nm, }) x y ( ( x, y) ) x= y= Figure 5. 3D and implicit plot for the Randic index for alpha=-/2 (2n+ 2m+ 4) (4n+ 4m 8) (6mn 5m 5n + 4) Rα = S S f = + +. α α α 4 6 9

11 of 8 Figure 6. 3D and implicit plot for the generalized Randic index for alpha=8 6. ( y x x y)( ) SSD( O{ nm, }) = S D + S D f ( x, y) = n + m - + 2mn. x= y= Figure 6. 3D and implicit plot for the Symmetric diision index 7. ( ) = ( f ) H O{ nm, } 2 SxJ ( x, y) x= = n+ m+ + 2 mn

12 2 of 8 Figure 7. 3D and implicit plot for the Harmonic index I ( O{ nm, }) = SxJDxDy ( f( x, y) ) - n- m 9 mn. x= = Figure 8. 3D and implicit plot for the Inerse Sum Index A( O{ n, m} ) = Sx Q 2 JDx Dy ( f( x, y) ) = - n- m+ + mn

13 3 of 8 Figure 9. 3D and implicit plot for the Augmented Zagreb index Theorem 3. Let O { nm, } be the octagonal network. Then the first and second Zagreb polynomials of O{ nm, } are M ( O x) = mnx mx nx + mx + nx + x + mx + 2nx 8x + x 牋? { nm, } M2 ( O{ nm, } x) = mnx mx nx + x + mx + nx x + mx + nx + x 2. 牋, Proof. Let O { nm, } be the octagonal network. Then. by the definition the first Zagreb polynomial (Definition 5) ( nm ) 牋 M? O x = x [ ] {, } = du+ d ( { nm, }) [ ] [ ] + + ( { nm, }) 2( { nm, ) 3 ( { nm} ) du+ d du+ d [ du+ d x x x ] u E O u E O } u E O, E( O{ nm, }) x E2( O{ nm, }) x E3 ( O{ nm, }) x 爘 = + + =6mnx u E O 5mx 5nx + 4mx + 4nx + 4x + 2mx + 2nx 8x + 4 x.

14 4 of 8 Figure 0. 3D implicit plot of first Zagreb polynomial and Interactie plot with two parameters 2. Now by definition the second Zagreb polynomial (Definition 6) ( { nm, } ) [ d d 牋? u M O x = x ] 2 = ( { nm, }) [ du d] [ du d] [ du d x x x ] + + ( { nm, }) ( { nm, }) ( { nm, }) u E O u E2 O u E3 O ( { nm, }) ( { nm, }) 3( { nm, }) 爘 = E O x + E? O x + E O x =6mnx u E O mx 5nx + 4x + 4mx + 4nx 8x + 2mx + 2nx + 4 x. Figure. 3D implicit plot of second Zagreb polynomial and Interactie plot with two parameters Proposition 2. Let O { nm, } be the octagonal network. Then the hyper-zagreb index, first multiple Zagreb index and the second multiple Zagreb index of O { nm, } are

15 5 of 8. ( nm ) HM? O{, } 26mn- 48m - 48n (? ) = PM O{ nm, } mn n m n m (? ) PM O nm 2 {, } mn n m n+ m =. Proof.. By definition of hyper-zagreb index (Definition 2) ( {, }) ( ) [ ] {, } HM O nm = du+ d u E O nm ( ) [ du d ] ( ) [ du d ] ( ) [ du d ] { nm, } 牋 { nm, } { nm, } ( { nm, }) ( { nm, }) ( { nm, }) ( 2n+ 2m+ ) ( n+ 4m ) = u E O u E2 O u E3 O 6 爘 25? 6? = E O + E2 O + E3 O = = 26mn- 48m- 48n+ 8. ( 6mn 5n 5m + 4) Figure 2. 3D and implicit plot for hyper-zagreb index 2. By the definition of first multiple Zagreb index (Definition 3) PM ( 燿 O ) ( ) [ d ] {, } ( ) [ du d ] ( ) [ du d ] ( ) [ du+ d ] 牋 { nm, } 2 { nm, } 3 牋 { nm, } ( { nm, }) E2( 牋 O ) E3( 牋 O?) { nm, } u u E O nm =4 = + = + + u E O u E O u E O E 牋 O { nm, } { nm, } = n+ 2m+ 4 4n+ 4m 8 6mn 5n 5m mn 6 5n 5m 0 4n+ 4m =

16 6 of 8 Figure 3. 3D and implicit plot for first multiple Zagreb index 3. By the definition of second multiple Zagreb index (Definition 4) PM2 ( 燿 O{, }) ( ) [ d ] {, } ( ) [ du d ] ( ) [ du d ] ( ) [ du d ] 牋 { nm, } 2 { nm, } 3 牋 { nm, } ( { nm, }) E2( 牋? O ) E3( 牋 O ) nm = u u E O nm = u E O u E O u E O E 牋 O { nm, } { nm, } = n+ 2m+ 4 4n+ 4m 8 6mn 5n 5m+ 4 =4 6 9 mn 0n 0m 4n+ 4m = Figure 4. 3D and implicit plot for second multiple Zagreb index

17 7 of 8 CONCLUSIONS AND DISCUSSION In this article we computed many topological indices for the octagonal network. At first, we gae the general closed form of M-polynomial of this family and recoer many degree-based topological indices out of it. We also computed multiple Zagreb indices and Zagreb polynomials of the octagonal network. We also gae graphs for the computed polynomials and topological indices see figure 2-4. Our graphical representations of topological indices show the dependency of the certain index on the structure. These results can play a ital role in determining properties of this network and its uses in industry, electronics, and pharmacy. Acknowledgments: This research is supported by Gyeongsang National Uniersity, Jinju 52828, Korea Author Contributions: Y. C. K. and W. N designed the problem and analysis tool; S. M. K. inestigated and produced the initial results; W. N. and M. M. erified and improed the final results and wrote the paper. Conflicts of Interest: The authors declare no conflict of interest." References. Stoer, D.; Normile, D. Buckytubes, Popular Science; Apr 92, 240(4) p3, The many faces of nanotech, Nature Nanotechnology;Oct2007, Vol. 2 Issue 0, p585, West, D. B. An Introduction to Graph Theory. (996). Prentice-Hall. 4. Deutsch, E.; Klazar, S. M-Polynomial and Degree-Based Topological Indices. Iranian Journal of Mathematical Chemistry, (205) 6(2), Gutman, I. Molecular graphs with minimal and maximal Randic indices. Croatica Chem. Acta (2002)75, Gutman, I. Degree-based topological indices. Croat. Chem. Acta (203)86, Vukiˇcei c, D. On the edge degrees of trees. Glas. Mat. Ser. III (2009)44(64) Gutman, I. Gutman, Some properties of the Wiener polynomials, Graph Theory Notes, New York 25 (993) Munir, M.; Nazeer, W.; Rafique, S.; Nizami, A. R.; Kang, S. M. M-polynomial and degree-based topological indices of Titania Nanotubes, Symmetry 206, 8, 7; doi:0.3390/sym Munir,M.; Nazeer,W.; Rafique, S.; Kang, S. M. M-polynomial and degree-based topological indices of Buckytubes,(Submitted). Kang, S.; Munir, M.; Nizami, A.; Shahzadi, Z.; Nazeer, W. Some Topological Inariants of the Möbius Ladder,Preprints 206, (doi: /preprints ). 2. Munir, M.; Nazeer, W.; Rafique, S.; Nizami, A.; Kang, S. M. Some Computational Aspects of Triangular Boron Nanotubes, Preprints 206, (doi: /preprints ). 3. Rucker, G.; Rucker, C. On topological indices, boiling points, and cycloalkanes. J. Chem. Inf. Comput. Sci. (999) 39, Klaz ar, S.; Gutman, I. A Comparison of the Schultz Molecular Topological Index with the Wiener Index. J. Chem. Inf. Comput.Sci. (996) 36, Deng H,Huang G, Jiang X A unified linear-programming modeling of some topological indices. J. Comb. Opt. (To appear) DOI 0.007/s Deng, H.; Yang, J. Xia F A general modeling of some ertex-degree based topological indices in benzenoid systems and phenylenes. Comp. Math. Appl. (20) 6, Gutman, I.; Trinajstic, N. Graph theory and molecular orbitals total _-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 972, 7,

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