Some Algebraic Polynomials and Topological Indices of Generalized Prism and Toroidal Polyhex Networks

Muhammad Ajmal 1, Waqas Nazeer 2, Mobeen Munir 2, Shin Min Kang 3,4 and Young Chel Kwun 5,* 1 Department of Mathematics, Government Muhammdan Anglo Orintal College, Lahore 54000, Pakistan; majmalkhan82@gmail.com 2 Division of Science and Technology, University of Education, Lahore 54000, Pakistan; nazeer.waqas@ue.edu.pk (W.N.); mmunir@ue.edu.pk (M.M.) 3 Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 52828, Korea; smkang@gnu.ac.kr or sm.kang@mail.cmuh.org.tw 4 Center for General Education, China Medical University, Taichung 40402, Taiwan 5 Department of Mathematics, Dong-A University, Busan 49315, Korea * Correspondence: yckwun@dau.ac.kr; Tel.: +82-51-200-7216


Introduction
Chemical reaction network theory is an area of applied mathematics that attempts to model the behavior of real world chemical systems.Since its foundation in the 1960s, 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 interesting problems that arise from the mathematical patterns in structures of material.
Cheminformatics is an emerging field in which quantitative structure-activity (QSAR) and Structure-property (QSPR) relationships predict the biological activities and properties of nanomaterial.In these studies, some physcio-chemical properties and topological indices are used to predict bioactivity of the chemical compounds [1][2][3][4][5].
The branch of chemistry which deals with the chemical structures with the help of mathematical tools is called mathematical chemistry.Chemical graph theory is the branch of mathematical chemistry that applies graph theory to mathematical modeling of chemical phenomena.In chemical graph theory, a molecular graph is a simple graph (having no loops and multiple edges) in which atoms and chemical bonds between them are represented by vertices and edges respectively.
A graph G with vertex set V(G) and edge set E(G) is connected if there exists a connection between any pair of vertices in G.The distance between two vertices u and v is denoted as d(u, v) and is the length of shortest path between u and v in graph G.The number of vertices of G, adjacent to a given Symmetry 2017, 9, 5 2 of 12 vertex v, is the "degree" of this vertex and will be denoted by d v .For details on basics of graph theory, any standard text such as [6] can be of great help.
A topological index is actually designed by transforming a chemical structure into a number.These topological indices associate certain physico-chemical properties like boiling point, stability, strain energy, etc. of chemical compounds.Graph theory has been of considerable use in this area of research.In 2013, Shirdel et al. [7] proposed the "hyper-Zagreb index", which is a degree-based index as In 2012, Ghorbani and Azimi [8] proposed two new variants of Zagreb indices; first multiple Zagreb index Recently in 2015, Furtula and Gutman [9] introduced another topological index called the forgotten index or F-index For more detail on the "F-index", we refer to the articles [10][11][12].The first Zagreb polynomial of G is defined as The second Zagreb polynomial of G is defined as The forgotten polynomial of a graph G is defined as In this article, we compute the first Zagreb polynomial, the second Zagreb polynomial and the forgotten polynomial of generalized prism and toroidal polyhex networks.We also compute some degree-based topological indices of these networks.Recently authors computed M-polynomials and related topological indices for Nanostar dendrimers [13], titania nanotubes [14], circulant graphs [15], polyhex nanotubes [16], and generalized prism and toroidal polyhex networks [17].The structures of Nanostar dendrimers, titania nanotubes, and polyhex nanotubes differ from fullerenes geometrically.Basic structural units of titania nanotubes are rectangles arranged differently for different types, whereas basic units of polyhex nanotubes are hexagons concatenated in different ways for different types.Nanostar dendrimers are macromolecules built in a tree-like structure.These materials have many applications in electronics, chemical processing, optics, and energy management and are used in flat panel display screens, hydrogen storage, robotics and artificial muscles, chemical sensors, and photography.Our results will help to determine physico-chemical properties like the heat of formation, strain energy, strength and fracture toughness of these materials.

Results and Discussions
In this part we give our main computational results.

Computational Aspects of Generalized Prism
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle or, in other words, some number of vertices connected in a closed chain.The cycle graph with n vertices is Symmetry 2017, 9, 5 3 of 12 called C n .The number of vertices in C n equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it.Figure 1 is a cycle of length 10.

Computational Aspects of Generalized Prism
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle or, in other words, some number of vertices connected in a closed chain.The cycle graph with n vertices is called .n C The number of vertices in n C equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it.Figure 1 is a cycle of length 10.
A path graph m P is a graph whose vertices can be listed in the order 12 ,? , ,?
ii vv  where  = 1,2, ⋯ ,  − 1 .Equivalently a path with at least two vertices is connected and has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2. Figure 2 shows a path graph 4 P .For a path m P , we have V (P m ) ={y j :1£ j £ m} and n m i j i j n j j i j i j E P x y x y i n j m x y x y j m x y x y i n j m For a cycle C n , we have A path graph P m is a graph whose vertices can be listed in the order v 1 , v 2 , • • •, v m such that the edges are {v i , v i+1 }, where i = 1, 2, • • • , m − 1. Equivalently a path with at least two vertices is connected and has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2. Figure 2 shows a path graph P 4 .

Computational Aspects of Generalized Prism
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle or, in other words, some number of vertices connected in a closed chain.The cycle graph with n vertices is called .n C The number of vertices in n C equals the number of edges, and every vertex has degree 2; that is, every vertex has exactly two edges incident with it.Figure 1 is a cycle of length 10.For a path m P , we have V (P m ) ={y j :1£ j £ m} and 1 ( ) { :1 1}.
The Cartesian product GH of graphs G and H is a graph such that the vertex set of GH is the Cartesian product ( ) ( )  For a path P m , we have The Cartesian product G H of graphs G and H is a graph such that the vertex set of G H is the Cartesian product V(G) × V(H) and any two vertices (u, u ) and (v, v ) are adjacent in G H if and only if either u = v and u is adjacent with v in H, or u = v and u is adjacent with v in G.
Let P {n,m} be the generalized prism graph that is obtained by Cartesian product of a cycle C n with a path P m as shown in Figure 3. Then V P {n,m} = (x i , y j    The generalized prism P {n,m} has been studied extensively in recent years.Kuo et al. [18] and Chiang et al. [19] studied distance-two labelings of P {n,m} .Siddiqui et al. [20] counted some topological indices of P {n,m} .Deming et al. [21] gave complete characterization of the Cartesian product of cycles and paths for their incidence chromatic numbers.Gravier et al. [22] showed the link between the existence of perfect Lee codes and minimum dominating sets of P {n,m} .Lai et al. [23] determined the edge addition number for the Cartesian product of a cycle with a path.Chang et al. [24] established upper bounds and lower bounds for global defensive alliance number of P {n,m} .
Theorem 1.Let P {n,m} be the generalized prism.Then the first Zagreb polynomial, second Zagreb polynomial, and forgotten polynomial of P {n,m} are = E 1 P {n,m} x 6 + E 2 P {n,m} x 7 + E 3 P {n,m} x 8 Following Figure 4 is the graph of first Zagreb polynomial P {7,8} .
Symmetry 2016, 9, 5 5 of 12 Following Figure 4 is the graph of first Zagreb polynomial {7,8} .P    2. Now, by definition, the second Zagreb polynomial is; = E 1 P {n,m} x 9 + E 2 P {n,m} x 12 + E 3 P {n,m} x 16 The graph of the second Zagreb polynomial of generalized prism is in Figure 5.
The graph of the forgotten polynomial of P {7,8} is given in Figure 6 below, Symmetry 2016, 9, 5 6 of 12 3.By definition, the forgotten polynomial is: The graph of the forgotten polynomial of {7,8} P is given in Figure 6 below,  

2.
By the definition of the first multiple Zagreb index, 3D plot of the multiple-Zagreb index of generalized prism is shown in Figure 8.

3.
By the definition of the second multiple Zagreb index, 3D plot for the second multiple-Zagreb index of generalized prism is given in Figure 9.
Symmetry 2016, 9, 5 8 of 12 3D plot for the second multiple-Zagreb index of generalized prism is given in Figure 9.
3D plot for the second multiple-Zagreb index of generalized prism is given in Figure 10.

4.
By the definition of the forgotten index, 5) .
3D plot for the second multiple-Zagreb index of generalized prism is given in Figure 10.
Symmetry 2016, 9, 5 8 of 12 16 3D plot for the second multiple-Zagreb index of generalized prism is given in Figure 9. 4. By the definition of the forgotten index, 324 625 36 .
3D plot for the second multiple-Zagreb index of generalized prism is given in Figure 10.

Computational Aspects of Toroidal Polyhex
The discovery of the fullerene molecules has stimulated many interests in other possibilities for carbons.Many properties of fullerenes can be studied using mathematical tools such as graph theory.A fullerene can be represented by a trivalent graph on a closed surface with pentagonal and hexagonal faces, such that its vertices are carbon atoms of the molecule.Two vertices are adjacent if there is a bond between corresponding atoms.In [25], authors considered fullerene's extension to other closed surfaces and showed that only four surfaces, sphere, torus, Klein bottle and projective (elliptic) plane, are possible.The spherical and elliptic fullerenes have 12 and 6 pentagons respectively.There are no pentagons in the toroidal's and the Klein bottle's fullerenes [26].
A toroidal fullerene (toroidal polyhex), obtained from 3D Polyhex Torus Figure 11, is a cubic bipartite graph embedded on the torus such that each face is a hexagon.The torus is a closed surface that can carry the graph of the toroidal polyhex in which all faces are hexagons and the degree of all vertices is 3.The optical and vibrational properties of toroidal carbon nanotubes can be found in [27].There have appeared a few works in the enumeration of perfect matchings of toroidal polyhexes by applying various techniques, such as transfer-matrix and permanent of the adjacency matrix.Ye et al. [28] have studied a k-resonance of toroidal polyhexes.Classification of all possible structures of fullerene Cayley graphs is given in [29] by Kang.The atom-bond connectivity index (ABC) and geometric-arithmetic index (GA) of the toroidal polyhex are computed in [30] by Baca et al.In [31], authors computed distance-based topological indices of eight infinite sequences of 3-generalized fullerenes.
In [32], authors presented a new extension of the generalized topological indices (GTI) approach to represent topological indices in a unified way.

Computational Aspects of Toroidal Polyhex
The discovery of the fullerene molecules has stimulated many interests in other possibilities for carbons.Many properties of fullerenes can be studied using mathematical tools such as graph theory.A fullerene can be represented by a trivalent graph on a closed surface with pentagonal and hexagonal faces, such that its vertices are carbon atoms of the molecule.Two vertices are adjacent if there is a bond between corresponding atoms.In [25], authors considered fullerene's extension to other closed surfaces and showed that only four surfaces, sphere, torus, Klein bottle and projective (elliptic) plane, are possible.The spherical and elliptic fullerenes have 12 and 6 pentagons respectively.There are no pentagons in the toroidal's and the Klein bottle's fullerenes [26].
A toroidal fullerene (toroidal polyhex), obtained from 3D Polyhex Torus Figure 11, is a cubic bipartite graph embedded on the torus such that each face is a hexagon.The torus is a closed surface that can carry the graph of the toroidal polyhex in which all faces are hexagons and the degree of all vertices is 3.The optical and vibrational properties of toroidal carbon nanotubes can be found in [27].There have appeared a few works in the enumeration of perfect matchings of toroidal polyhexes by applying various techniques, such as transfer-matrix and permanent of the adjacency matrix.Ye et al. [28] have studied a k-resonance of toroidal polyhexes.Classification of all possible structures of fullerene Cayley graphs is given in [29] by Kang.The atom-bond connectivity index (ABC) and geometric-arithmetic index (GA) of the toroidal polyhex are computed in [30] by Baca et al.In [31], authors computed distance-based topological indices of eight infinite sequences of 3-generalized fullerenes.
In [32], authors presented a new extension of the generalized topological indices (GTI) approach to represent topological indices in a unified way.The set of vertices of the toroidal polyhex is: The set of edges of the toroidal polyhex is splitted into mutually disjointed subsets; such that for even i , such that 0 2, in Let L be a regular hexagonal lattice and n P m be a m × n quadrilateral section (with m hexagons on the top and bottom sides and n hexagons on the lateral sides; n is even), cut from the regular hexagonal lattice L. First identify two lateral sides of n P m to form a cylinder, and finally identify the top and bottom sides of n P m at their corresponding points.From this we get a toroidal polyhex H {m,n} with mn hexagons.
The set of vertices of the toroidal polyhex is: The set of edges of the toroidal polyhex is splitted into mutually disjointed subsets; such that for even i, such that 0 ≤ i ≤ n − 2, we have A i = u i j v i j : 0 ≤ j ≤ m − 1 and A i = v i j u i j+1 : 0 ≤ j ≤ m − 1 , and for i odd and 1 ≤ i ≤ n − 1, we have and for 0 ≤ i ≤ n − 1 we have C i = v i j u i+1 j : 0 ≤ j ≤ m − 1 , where i is taken modulo n and j is taken modulo m.
We can easily observe from Figure 12 that the number of vertices in H {m,n} are 2mn and the number of edges in H {m,n} are 3mn.Note that there is only one type of edge in a toroidal polyhex, based on degrees of end vertices of each edge.

Conclusions and Discussion
In this article, we computed closed forms of topological indices for a generalized prism and toroidal polyhex.We also gave closed forms of some well-known polynomials concerning these structures.Further, we gave graphs (Figures 4-10) for the computed polynomials and topological indices to correlate these indices with involved parameters of structure.These results can play a vital role in industry and pharmacy.It is important to remark that the methodology described above can be employed in recently developed nanomaterials, nanotubes, and polymers [33][34][35].Some examples are Boron nanotubes, aluminosilicate/aluminugerminate, SiO 2 -layered structure and titania nanotubes.
The Cartesian product GH of graphs G and H is a graph such that the vertex set of GH is the Cartesian product ( ) ( )V G V H  and any two vertices   , ' ?uu and   , ' ?vv are adjacent in GH if and only if either u = v and '? u is adjacent with v' in H, or u' = v' and u is adjacent with v in G. Let { , } nm P be the generalized prism graph that is obtained by Cartesian product of a cycle


and any two vertices   , ' ?uu and   , ' ?vv are adjacent in GH if and only if either u = v and '? u is adjacent with v' in H, or u' = v' and u is adjacent with v in G. Let { , } nm P be the generalized prism graph that is obtained by Cartesian product of a cycle n C with a path m P as shown in Figure 3

Figure 4 . 2 .
Figure 4. Graph of the first Zagreb polynomial of {7,8} P the second Zagreb polynomial of generalized prism is in Figure5.

Following Figure 4 Figure 4 .. 2 .
Figure 4. Graph of the first Zagreb polynomial of {7,8} P the second Zagreb polynomial of generalized prism is in Figure5.
the multiple-Zagreb index of generalized prism is shown in Figure8.

Figure 8 .
Figure 8. 3D plot for the first multiple-Zagreb index.
the multiple-Zagreb index of generalized prism is shown in Figure8.

Figure 9 .
Figure 9. 3D plot for the second multiple-Zagreb index.4. By the definition of the forgotten index,

Figure 9 .
Figure 9. 3D plot for the second multiple-Zagreb index.

Figure 9 .
Figure 9. 3D plot for the second multiple-Zagreb index.

Figure 10 .
Figure 10.3D plot of the forgotten index.

Figure 10 .
Figure 10.3D plot of the forgotten index.

Proposition 2 .
Let { , } mn H be the toroidal polyhex.Then the hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb inde,x and forgotten index of { , }
The edge partition E 1 (H {m,n} ) contains 3mn edges uv, where d u = d v = 3.Let H {m,n} be the toroidal polyhex.Then the first Zagreb polynomial, second Zagreb polynomial, and forgotten polynomial of H {m,n} are;M 1 H {m,n} , x = 3mnx 6 , M 2 H {m,n} , x = 3mnx9, and F H {m,n} , x = mnx18.Let H {m,n} be the toroidal polyhex.Then the hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb inde, x and forgotten index of H {m,n} are; HM H {m,n} = 108 mn , PM 1 H {m,n} = 216 mn , PM 2 H {m,n} = 729 mn , F H {m,n} = 5832 mn .Note that there is only one type of edge in a toroidal polyhex, based on degrees of end vertices of each edge.The edge partition     where i is taken modulo n and j is taken modulo m. uv dd 