SS symmetry Article
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 2 3 4 5
*
Department of Mathematics, Government Muhammdan Anglo Orintal College, Lahore 54000, Pakistan;
[email protected] Division of Science and Technology, University of Education, Lahore 54000, Pakistan;
[email protected] (W.N.);
[email protected] (M.M.) Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University, Jinju 52828, Korea;
[email protected] or
[email protected] Center for General Education, China Medical University, Taichung 40402, Taiwan Department of Mathematics, Dong-A University, Busan 49315, Korea Correspondence:
[email protected]; Tel.: +82-51-200-7216
Academic Editor: Angel Garrido Received: 1 December 2016; Accepted: 20 December 2016; Published: 29 December 2016
Abstract: A topological index of graph G is a numerical parameter related to G, which characterizes its topology and is preserved under isomorphism of graphs. Properties of the chemical compounds and topological indices are correlated. In this report, we computed closed forms of first Zagreb, second Zagreb, and forgotten polynomials of generalized prism and toroidal polyhex networks. We also computed hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb index, and forgotten index of these networks. Moreover we gave graphical representation of our results, showing the technical dependence of each topological index and polynomial on the involved structural parameters. Keywords: forgotten polynomial; Zagreb polynomial; topological index; network
1. 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 [1–5]. 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
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vertex v, is the “degree” of this vertex and will be denoted by dv . 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 HM ( G ) = ∑ [du + dv ]2 uv∈ E( G )
In 2012, Ghorbani and Azimi [8] proposed two new variants of Zagreb indices; first multiple Zagreb index PM1 (G) = ∏ [du + dv ] and second multiple Zagreb index PM2 (G) = ∏ [du × dv ]. uv∈E(G)
uv∈E(G)
Recently in 2015, Furtula and Gutman [9] introduced another topological index called the forgotten index or F-index h i F(G) = ∑ (du )2 + (dv )2 . uv∈E(G)
For more detail on the “F-index”, we refer to the articles [10–12]. The first Zagreb polynomial of G is defined as
∑
M1 (G, x) =
x [du +dv ] .
uv∈E(G)
The second Zagreb polynomial of G is defined as M2 (G, x) =
∑
x [du ×dv ] .
uv∈E(G)
The forgotten polynomial of a graph G is defined as F(G, x) =
∑
2
x[(du )
+(dv )2 ]
.
uv∈E(G)
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. 2. Results and Discussions In this part we give our main computational results. 2.1. 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
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2.1. Computational Aspects of Generalized Prism 2.1. Computational Aspects of Generalized Prism Symmetry 2017, 9, 5theory, a cycle graph or circular graph is a graph that consists of a single cycle 3or, of 12 In graph in 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 other words, some number of vertices connected in a closed chain. The cycle graph with n vertices is called Cn . The number of vertices in Cn equals the number of edges, and every vertex has degree 2; The number number of vertices in CCnn equals equalsthe thenumber numberof ofedges, edges,and and every every vertex vertex has degree 2; called C Cnn.. The that is, every vertex has exactly two edges incident with it. Figure 1 is a cycle of length 10. that is, every vertex has exactly two edges incident with it. Figure 1 is a cycle of length incident with it. cycle of length 10. 10.
Figure 1. Cycle graph C10 Figure 10. Figure 1. 1. Cycle Cycle graph graph CC10
For a cycle Cn , we have V (Cn ) {xi :1 i n}, and E(Cn ) {xi xi 1 :1 i n 1} {xn x1}. Cn,, we (Cn) )= {{xxi :1 i n}, and E(Cn ) {xi xi 1 :1 i n 1}≤n{− xn x1}}.∪ {x x }. For aa cycle cycle C wehave haveVV(C For ≤ n n n 1 i : 1 ≤ i ≤ n}, and E(Cn ) = { xi xi+1 : v1 ,? v i, ,?vm 1such A path graph Pm is a graph whose vertices can be listed in the order v11,?v22, ,?vm that the A path graph P is a graph whose vertices can be listed in the order v , v , · · · , v such m A path graph Pmm is a graph whose vertices can be listed in the order 1 2 such thatthat the { v ,? v }, edges are where 𝑖 = 1,2, ⋯ , 𝑚 − 1 . Equivalently a path with at least two vertices is i i 1 the edges are v , v , where i = 1, 2, · · · , m − 1. Equivalently a path with at least two vertices { } i i + 1 edges are {vi ,?vi 1}, where 𝑖 = 1,2, ⋯ , 𝑚 − 1 . Equivalently a path with at least two vertices is is any) have have connected and has two terminal vertices (vertices that have degree 1), while all others (if any) connected and has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2. 2. Figure Figure 22 shows shows aa path path graph graph P4P.4 . degree degree 2. Figure 2 shows a path graph P4 .
Figure Figure 2. 2. Path Pathgraph graph PP44. . Figure 2. Path graph P4 .
For aa path path PP , wehave haveV (VP(P {x j xxjjx1j+ :11 : 1j ≤nj≤ 1}.n − 1 . ) ={y m}m and For , we y jj ::1£ 1 ≤j j£≤ andEE( P ( nP)n ) = mm m )m= For a path P , we have and E ( P ) { x x :1 j n 1}. V (P ) ={y :1£ j £ m} m n j j 1 m j The and HH is aisgraph such thatthat the vertex set ofset G H ofofgraphs G isHthe The Cartesian Cartesianproduct productGG H graphsG G and a graph such the vertex of H is 0 0 G H G H The Cartesian product of graphs G and H is a graph such that the vertex set of is Cartesian product V ( G ) × V ( H ) and any two vertices u, u and v, v are adjacent in G H if and ( ) ( ) the Cartesian product V (G) V ( H ) and any two vertices u, u '? and v, v '?are adjacent in G H if u, u 'u?isand V ( H ) and the Cartesian are adjacent v, v '?with only if either u product = v and V u0(G is )adjacent withany v0 intwo H, orvertices u0 = v0 and adjacent v in G. in G H if u '? is adjacent with v’ in H, or u’ = v’ and u is adjacent with v in G. and only if either u = v and Let P be the generalized prism graph that is obtained by Cartesian product u '? } and only if{n,m either u = v and is adjacent with v’ in H, or u’ u is adjacent with v inofG.a cycle = v’and Cn Let P{n,m} be the generalized prism graph that is obtained by Cartesian product of a cycle Cn Let aP{path be generalized prism3.graph obtained ofj a≤cycle Cn with as shown in Figure Thenthat V Pis{n,m ( xiCartesian , y j ) : 1 ≤ product i ≤ n, 1 ≤ m and m the n,m} P } = by a path P as shown in Figure 3. Then V P with m {n,m} ( xi ,y j ) :1 i n,1 j m and E Pa{n,m , yFigure i≤ n − 1, m V 1P{≤n,mj }≤ m ( x∪i , y(j x) n:1, yj )( i x1n,,1y j) :j 1 ≤ mj≤ with path shown Then and i, y j )( xi +1in j ) : 1 ≤3. } =Pm ( xas E P{n,m} ( xi , y j )(∪ xi 1(,xyi , jy)j )( :1xi , iy m1 1 ≤ i j≤ n, ≤ j( x≤n ,my − j +n 1 ):1,1 j )(1x1., y j ) :1 j m
E P{n,m} ( xi , y j )( xi 1 , y j ) :1 i n 1,1 j m ( xn , y j )( x1, y j ) :1 j m ( x , y )( x , y ) :1 i n,1 j m 1 . ( xii , y jj )( xii , y jj11) :1 i n,1 j m 1 .
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Figure 3. The generalized prism P{n,m} . Figure 3. The generalized prism P{n,m} .
The generalized generalized prism prism PP{n,m} has hasbeen beenstudied studiedextensively extensively in inrecent recentyears. years. Kuo Kuo et et al. al. [18] [18] and and The {n,m} Chiang etetal.al. [19] studied distance-two labelings of} . P al. [20]some counted some Chiang [19] studied distance-two labelings of P{n,m Siddiqui et al. [20]etcounted topological {n,m} . Siddiqui indices of P . Deming et al. [21] gave complete characterization of the Cartesian product of cycles } topological{n,m indices of P{n,m}. Deming et al. [21] gave complete characterization of the Cartesian and paths for their incidence chromatic numbers. Gravier et al. [22] showed the link between the product of cycles and paths for their incidence chromatic numbers. Gravier et al. [22] showed the existence of perfect Lee codes and minimum dominating sets of P{n,m} . Lai et al. [23] determined the link between the existence of perfect Lee codes and minimum dominating sets of P{n,m} . Lai et al. edge addition number for the Cartesian product of a cycle with a path. Chang et al. [24] established [23] determined the edge addition number for the Cartesian product of a cycle with a path. Chang et upper bounds and lower bounds for global defensive alliance number of P{n,m} . 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 P{n,m Theorem 1. polynomial Let P{n,m}of be the} are generalized prism. Then the first Zagreb polynomial, second Zagreb polynomial, and forgotten polynomial of P n,m}7 are 6 1. M1 P{n,m} , x = 2mnx8 − 5nx8 + {2nx + 2nx , 12 + 2nx9 , 2. M2 P{n,m} , x = 2mnx168 − 5nx816 + 2nx 7 6 M1 P{n, m} , x 2mnx 5nx 2nx 2nx , 3. 1.F P{n,m} , x = 2mnx32 − 5nx32 + 2nx25 + 2nx18 .
M 2 P{n,m} , x 2mnx16 5nx16 2nx12 2nx9 ,
2. Let P Proof. {n,m} be the generalized prism with defining parameters m and n. The number of vertices and edges in generalized prism P n,m} are25nm and n(2m − 1) respectively. There are three types of F P{n,m} , x 2mnx32 5nx{32 2nx 2nx18 . edges 3. in P{n,m} based on degrees of end vertices of each edge. The first edge partition E1 ( P{n,m} ) contains 2n edges uv, where du = dv = 3. The second edge partition E2 ( P{n,m} ) contains 2n edges Proof. Let duP{n=,m}3, be prism with defining m and n. edges The number of uv, where dv the = 4.generalized The third edge partition E3 ( P{n,m}parameters ) contains 2mn − 5n uv, where dvertices 4. Then; and edges in generalized prism P{n,m} are nm and n(2m − 1) respectively. There are three u = dv =
types of edges in P{n,m} based on degrees of end vertices of each edge. The first edge partition E1 ( P{n,m} ) contains 2n edges uv, where du dv = 3 . The second edge partition E2 ( P{n,m} )
contains 2n edges uv, where du 3, dv 4. The third edge partition E3 ( P{n,m} ) contains
2mn 5n edges uv, where du dv 4. Then; 1.
by definition, the first Zagreb polynomial is;
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d d by definition, the first M , x Zagreb xx duu dvv is; polynomial M11 P P{{nn,,m m}} , x uv [du +dv ] uv E E P P{{nn,, m m}} = M1 P{n,m} , x x ∑ d d d d du ddv E( P{n,m}x) d xx duu dvv xxduu dvv uv∈ x u v v] + = x [du +dv ]uv x [du +duv x [du +dv ] uv ∑ m}} m}} uv E E11 P P{{∑ uv+ E E22 P P{{nn,,∑ uv E E33 P P{{nn ,, m nn,, m m m}} uv∈ E1 ( P{n,m} ) uv∈ E2 ( P{n,m} ) uv∈ E3 ( P{n,m} ) 8 7 8 m} xx8 3 7 P {n,m} xx7 E P || xx66 P E 62+? 11 E } { n , || E E= P{{1nn,,m E ? P E P P + P P E x E x m}{n,m} 2 {n,2m} {n,m} 3 {n, m}3 {n,m} x
7 6 88 8 −885nx8 + 2mnx 2= mnx 5nx 2nx77 2nx 2nx66+. 2nx .
2mnx 5nx 2nx 2nx .
P P{7,8}P.. Following Figurethe 4 is the graph of first Zagreb polynomial . Following Figure Following Figure 44 is is the graph graph of of first first Zagreb Zagreb polynomial polynomial {7,8} {7,8}
Figure 4. Graph of the first Zagreb polynomial of . Figure 4. 4. Graph Graph of of the the first first Zagreb Zagreb polynomial polynomial of of PP P{{7,8} Figure {7,8} 7,8} . .
2. 2. 2.
Now, by definition, the second Zagreb polynomial is; Now, by by definition, definition, the the second second Zagreb Zagrebpolynomial polynomialis; is; Now, d d M , x x duu dvv M2 P P{{nn,,m m}}, ,xx = ∑ x x [du ×dv ] M2 2P{n,m } uv E } ) ,, m uv uv E ∈P PE{{nn( P m{}}n,m
d [ddu ×dv ] + d d[du ×dv ] ∑ x dduu ddxvv[du ×dv ] = ∑ xx dxuu dvv ∑ xxd}uu) xdvv + uv∈ E1 ( P{n,m} ) uv∈ E2 ( P{n,m uv∈ E3 ( Px ) EE1 uv E2 PP{n,m} EE3 PP{n uv P m}} 9 uv {n,m} uv uv E uv 1 P{{nn,, m 2 {n , m} 12 3 {n ,, m m }} 16 = E1 P{n,m} x + E2 P{n,m} x + E3 P{n,m} x 9 12 12 E9 P 16 | E P x16 16 | x 12 x nn,, m | E P{{nn,,m x95nx E E1622 +??P P{{2nx E33. P{{nn,,m =11 2mnx + 2nx m}} |− m}} x m}} x
16 5nx16
12 2nx12
2mnx 5nx 2nx 2nx . The graph of the second Zagreb polynomial of generalized prism is in Figure 5. The The graph graph of of the the second second Zagreb Zagreb polynomial polynomial of of generalized generalized prism prism is is in in Figure Figure 5. 5. 16 2mnx16
2nx99 .
P{7,8} Figure {7,8} . . Figure 5. 5. Graph Graph of of the the second second Zagreb Zagreb polynomial polynomial of of P P {7,8}.
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3. By definition, the forgotten polynomial is: 3. By definition, the forgotten polynomial is:
F P{n,m} , x F P{n,m = }, x
∑
d 2 d 2 u v
x
{n,m} )
uvE P{n, m} uv∈ E( P
x [(du )
2
+(dv )2 ]
2 2
2
2 2
2
2
d xd[( du dv x [(du ) v du) +(dv ) ] + ∑ x dux[(dud)v +( dv ) ] + ∑ u ∑ uv∈ x x E1 ( P{n,m} ) uv∈ E2 ( P{n,m} ) uv∈ E3 ( P{n,m} ) uvE E2 P{n,m uv E1 P{n,m} 18 uv } 25 3 P{n32 , m} = E1 P{n,m} x + E2 P{n,m} x + E3 P{n,m} x
=
2
2
2
2
+(dv )2 ]
| E1 P{n,m}32| x18 E32 ?P ,m} 25 x 25 E3 18 P x32 = 2mnx − 5nx 2 +{n2nx + 2nx {.n,m} 2mnx32 5nx32 2nx 25 2nx18 .
The graph the forgotten polynomial givenininFigure Figure6 6below, below, {7,8}isisgiven The graph ofof the forgotten polynomial ofofPP {7,8}
Figure 6. 6. Graph of of thethe forgotten polynomial ofof P{7,8} Figure Graph forgotten polynomial P{7,8.} .
Proposition the generalized generalizedprism. prism.Then Thenthethe hyper-Zagreb index, multiple Zagreb Proposition1.1.Let Let P{{nn,m hyper-Zagreb index, firstfirst multiple Zagreb index, ,m}} be the second multiple Zagreb index, and forgotten index of P are index, second multiple Zagreb index, and forgotten index{n,m of }P{n,m} are 1.1. HM 150n 128mn, HMP{nP,m } {n,m } = −150n + 128mn, 2n 2n2n 2mn 5n2mn−5n 2n 2.2. PM PM , 1 1P{nP,m } =6 {n,m } = 67 ×78 × 8 , mn mn 4m 4n+4m −m 10n−10m PM = 531441 12 , O{O 310×n310 124n× , 3.3. PM } 31441 2 2? n,{mn,m } = n(2m−5) . n n +n625n n m-5) = 324 +(236 . 4.4. F F P{Pn{,mn,m } } 324 625 36
Proof. Proof.
1.1. ByBy definition ofof the hyper-Zagreb index, definition the hyper-Zagreb index,
HM P{n,m} HM P{n,m}
=
2 d d ∑ u [dvu + dv ]
2
[ d 2 + d ]2 + [ d 2 + d ]2 + [d2u + dv ]2 d) v u v uv∈E ∑ d d uv∈ E1∑(Pd{un,m} d) v u v uv∈E2∑(dP{u u v n,m} uvE P 3 ( P{n,m} ) uvE1 P{n, m } uv E P 2 {n,m} 3 {n, m } = 36 E P +49 E P + 64 E P uvE Puv {n∈ , mE}( P{n,m} )
=
1
{n,m}
2
{n,m}
36 | E= 49 64−E5n 36 2n 49(E2n + ){n,m} ? 1 P {n(, m } ?) |+ 2 )?P {n,64 m}(2mn 3 P
3
{n,m}
150n 36 2n= −49 64 2mn 5n 2n + 128mn.
150n 128mn. The Figure 7 is 3D plot of the Hyper Zagreb index of generalized prism. The Figure 7 is 3D plot of the Hyper Zagreb index of generalized prism.
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Figure 3D plot for the hyper-Zagreb index. Figure 3D plot for the hyper-Zagreb index. Figure 7.7.7. 3D plot for the hyper-Zagreb index.
By thedefinition definition of thefirst first multipleZagreb Zagreb index, 2.2.2. By Bythe the definitionofofthe the firstmultiple multiple Zagrebindex, index,
PM 1 1P{P } PM dudu dvdv n,}m n{, m PM1 P{n,m} = uv E P{n,m} [du + dv ] uvE P ∏ {n,m}
uv∈ E( P{n,m} )
. 8 8 × 7 7×7 8
6
=
+ dv ]
]× d d udu dvd[dv u v d ∏ d udu[dduvd+ ∏ d udu d[d ∏ v uv+ dv ] × v uv E P uv E P uv E P uv ∈ E ( P ) uv ∈ E ( P ) uv ∈ E ( n,}m{}n,m} n,2}m}{n,m} n,}m3} P{n,m} ) uvE1 1P{n{,1m uvE2 2P{n{,m uvE3 3P{n{, m | E1 ( P{n,m} )| | E2 ( P{n,m} )| | E3 ( P{n,m} )| = 6 ?P ×7 P ×8 2P{n{, m n,}m} n,}m} E3E3P{nP{, m n,}m|}2n| E2E2mn | E1| E1?P2n {n{,m −5n
6= 6
2 mn 5n 2 n2 n 27n2 n 28mn =6=6 7 8 5 n. . 3D plot of the multiple-Zagreb index of generalized prism is shown in Figure 8. 3Dplot plotofofthe themultiple-Zagreb multiple-Zagrebindex indexofofgeneralized generalizedprism prismisisshown shownininFigure Figure8.8. 3D
Figure 8. 3D plot for the first multiple-Zagreb index. Figure Figure8.8.3D 3Dplot plotfor forthe thefirst firstmultiple-Zagreb multiple-Zagrebindex. index.
thedefinition definitionofofthe thesecond secondmultiple multipleZagreb Zagrebindex, index, 3.3. ByBythe
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By the definition of the second multiple Zagreb index, Symmetry 2016, 9, 5
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PM2 P{n,m}
=
∏
[du × dv ]
8 of 12 × [du × dv ] × [du × dv ] ∏ ∏ uv∈ E1 ( P{n,m} ) uv∈ E2 ( P{n,m} ) du d|vE (P d|Eu (dPv )| uvd∈uE3d(Pv {n,m} ) | E ( P )| )| PM 2 P{= d d 2 3 {n,m} 1 { n,m } { n,m } n,m}9 uvE P × 12 u v uvE P × 16 uvE3 P{n, m} 1 {n, m} 2 {n, m} uvE P2n {n, m} 2n 2mn − 5n = 9 |E1× 12 × 16 . E2 P{n,m} E3 P{n,m} ?P{n,m} | 12 9 16 du d v d u d v d u d v n E1 P uvE2 P{n, m} uvE3 P{n, m} } 2 mn 5n. =92uv 122{nn,m16 3D plot for the second multiple-Zagreb index of generalized prism is given in Figure 9. E2 P{n,m} E3 P{n,m} | E1 ?P{n,m} | 3D plot for the second generalized prism is given in Figure 9. 9 multiple-Zagreb 12 index of 16 PM 2 P{n,m}uv∈ E( P{ n,m} )
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=
du d v uv E P{n, m}[du × dv ] ∏
=92n 122n 162mn 5n.
3D plot for the second multiple-Zagreb index of generalized prism is given in Figure 9.
Figure 9. 3D plot for the second multiple-Zagreb index.
Figure 9. 3D plot for the second multiple-Zagreb index. 4.
4.
By the definition of the forgotten index,
Figure plot By the definition of the forgotten index, 2 for the second multiple-Zagreb index. d 9.2 3D F P d
{n,m}
u
v
uvE P{n, mh} definition i 4. By the of the forgotten index, 2 2 2 (dv )2 F P{n,m} = d + ( ) ∏ u d d
2 2 2 2 } ) u 2 du dv du dv v2 ∈ E( P{n,m F P{n,muv h h} du dv i uvE2 P{n,m} h uvE3 Pi } uvE 1 P{n,m {n, m} 2 2 2 2
( ) + ( ) × (du ) + (dv ) × ( d u )2 + ( d v )2 ∏ uv∈ E3 ( 2P{n,m} ) 2 dv 2 du dv = × × uvE3 P{n, m} n + 625n + 36n(2m−5) . = 324 E2 P{n, m} index E3 P{of 3D plot for the second prism is given in Figure 10. | E1 ?P{nmultiple-Zagreb n, m}generalized , m} | 18 25 31 =
uv E P{n, m} d u d ∏ ∏ E2 P{n,vm} E3 P{n, m} | E1 ?P{n, m} | uv∈E18 ) 1 ( P{n,m} ) 2 2 31 uv∈ E2 ( P{n,m 25 } 2 d d v )| ( P{nn,m} )| n |unE(22 m ( P-5) | E3 ( P{n,m }d)|u {n,m } 18| E1324 25 31 625 36 . uvE1 P{n,m} uvE2 P{n,m}
324n 625n 36n(2m-5).
3D plot for the second multiple-Zagreb index of generalized prism is given in Figure 10. 3D plot for the second multiple-Zagreb index of generalized prism is given in Figure 10.
Figure 10. 3D plot of the forgotten index.
i
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Figure 10. 3D plot of the forgotten index.
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2.2. Computational Aspects of Toroidal Polyhex 2.2. Computational Aspects of Toroidal Polyhex The discovery of the fullerene molecules has stimulated many interests in other possibilities for The of the fullerene molecules stimulated many interests in other for carbons. discovery Many properties of fullerenes can has be studied using mathematical tools possibilities such as graph carbons. Many properties of fullerenes can be studied using mathematical tools such as graph theory. theory. A fullerene can be represented by a trivalent graph on a closed surface with pentagonal and A fullerene faces, can besuch represented by a trivalent graphatoms on a closed withTwo pentagonal hexagonal hexagonal that its vertices are carbon of thesurface molecule. vertices and are adjacent if faces, such that its vertices are carbon atoms of the molecule. Two vertices are adjacent if there is a there is a bond between corresponding atoms. In [25], authors considered fullerene’s extension to bond between corresponding atoms. In [25], authors considered fullerene’s extension to other closed other closed surfaces and showed that only four surfaces, sphere, torus, Klein bottle and projective surfaces showed only four sphere, bottle and projective plane, (elliptic) and plane, are that possible. Thesurfaces, spherical and torus, ellipticKlein fullerenes have 12 and(elliptic) 6 pentagons are possible. The spherical and elliptic fullerenes have 12 and 6 pentagons respectively. There are no respectively. There are no pentagons in the toroidal’s and the Klein bottle’s fullerenes [26]. 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 A toroidal fullerene (toroidal from 3D face Polyhex FigureThe 11, is a cubic bipartite graph embedded on polyhex), the torusobtained such that each is aTorus hexagon. torus is abipartite closed graph embedded on the torus such that each face is a hexagon. The torus is a closed surface that surface that can carry the graph of the toroidal polyhex in which all faces are hexagons and can the carry the of the is toroidal in which all faces are hexagons and thecarbon degreenanotubes of all vertices 3. degree ofgraph all vertices 3. The polyhex optical and vibrational properties of toroidal canisbe The optical and vibrational properties of toroidal carbon nanotubes can be found in [27]. There have found in [27]. There have appeared a few works in the enumeration of perfect matchings of toroidal appeared few works in the enumeration perfect matchings of and toroidal polyhexes byadjacency applying polyhexesaby applying various techniques, of such as transfer-matrix permanent of the various techniques, transfer-matrix and permanent ofpolyhexes. the adjacency matrix. Ye et have matrix. Ye et al. [28]such haveasstudied a k-resonance of toroidal Classification ofal. all[28] possible studied a k-resonance of toroidal polyhexes. Classification of all possible structures of fullerene Cayley structures of fullerene Cayley graphs is given in [29] by Kang. The atom-bond connectivity index graphs is given in [29] by Kang. The atom-bond connectivity index (ABC) and geometric–arithmetic (ABC) and geometric–arithmetic index (GA) of the toroidal polyhex are computed in [30] by Baca et index the toroidal polyhex are computed in [30] byindices Baca etof al.eight In [31], authors computed al. In (GA) [31], of authors computed distance-based topological infinite sequences of distance-based topological indices of eight infinite sequences of 3-generalized fullerenes. 3-generalized fullerenes. In a new extension of theof generalized topological indices (GTI) approach In [32], [32],authors authorspresented presented a new extension the generalized topological indices (GTI) to represent topological indices in a unified way. approach to represent topological indices in a unified way.
Figure 11. Polyhex Torus. Figure 11. Polyhex Torus.
Let L be a regular hexagonal lattice and n Pm be a m n quadrilateral section (with m Let L be a regular hexagonal lattice and n Pm be a m × n quadrilateral section (with m hexagons on hexagons on the top and bottom sides and n hexagons on the sides; n is even), cuthexagonal from the the top and bottom sides and n hexagons on the lateral sides; n is lateral even), cut from the regular n n P lateral regularL.hexagonal lattice L. lateral First identify sides of Pm and to form a cylinder, andtop finally lattice First identify two sides oftwo finally identify the and m to form a cylinder, n n bottom sides of P at their corresponding points. From this we get a toroidal polyhex H with identify the top and toroidal Pm at their corresponding points. From this we get {am,n m bottom sides of } mn hexagons. polyhex H{m,n} with mn hexagons. The set of vertices of the toroidal polyhex is: The set of vertices of the toroidal polyhex is: n o i ii i V ( H{Vm,n ) = v , v : 0 ≤ i ≤ n − 1, 0 ≤ j ≤ m − 1 . } ( H{m,n} ) j{v jj, v j : 0 i n 1,0 j m 1}. The set of edges of the toroidal polyhex into mutually disjointed subsets; such that foro even n is splitted o mutually ndisjointed The set of edges of the toroidal polyhex is splitted into subsets; such that i i 0 i i, such that 0 ≤ i ≤ n − 2, we have Ai = u j v j : 0 ≤ ji ≤i m − 1 and A i = v j uij+1 : 0i ≤i j ≤ m − 1 , and 'i {v j u j 1 : 0 j m 1} for even i ,such that 0 i n 2, we have n Ai {u j v j : 0 j mo 1} and An o for i odd and 1 ≤ i ≤ n − 1, we have Bi = vij uij : 0 ≤ j ≤ m − 1 and B0 i = uij vij+1 : 0 ≤ j ≤ m − 1 ,,
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and for i odd and 1 i n 1, we have Bi {vij uij : 0 j m 1} and B 'i {uij vij 1 : 0 j m 1} , Symmetry 2017, 9, 5 of 12 and for 0 i n 1 we have Ci {vij uij1 : 0 j m 1}, where i is taken modulo n and j is 10 taken
modulo m. n o and for 0 ≤ i ≤ n − 1 we nhave Ci = vij uij+1 : 0 ≤ j ≤ m − 1 , where i is taken modulo n and j is taken 1 n 1 2 modulo m. E ( H Hence Ci . We can easily observe from Figure 12 {m,n} ) n A2i A '2i B2i 1 B '2i 1 2 −1
n−1
0 0 ( A ∪ A0 ∪ B i 0 Hence E( H{m,n} ) = i ∪ 2i 2i 2i+1 ∪ B 2i+1 ) ∪ Ci . We can easily observe from Figure 12 i=0in H i= 0 that the number of vertices are 2mn and the number of edges in H ,n} are 3mn. Note {m,n}are 2mn and the number that the number of vertices in H{m,n of edges in H{m,n{}mare 3mn. Note that } that is only one type of edge in a toroidal polyhex, on degrees ofvertices end vertices of edge. each therethere is only one type of edge in a toroidal polyhex, basedbased on degrees of end of each edge. Thepartition edge partition 3mnuv, edges uv,dwhere E1}()Hcontains du3. dv = 3. The edge E1 ( H{m,n 3mn edges where u = dv = {m,n} ) contains
Let HH{m,n} bebethe Theorem 2. 2. Let Theorem thetoroidal toroidalpolyhex. polyhex.Then Thenthe thefirst firstZagreb Zagrebpolynomial, polynomial, second second Zagreb Zagreb polynomial, polynomial, {m,n} and forgotten polynomial of H{m,n} are; and forgotten polynomial of H{m,n} are; 6 6, M x 3mnx M11 H{{m , x =3mnx , m,n } ,n} 9 9 , and M , x = 3mnx M22 H H{{m,n m,n}} ,x 3mnx , and
18 F H{m,n} , x = mnx 18 .
F H{m,n} , x mnx .
Proposition 2. Let Let HH toroidal polyhex. Then the hyper-Zagreb index, first multiple thethe toroidal polyhex. Then the hyper-Zagreb index, first multiple ZagrebZagreb index, {m,n {m,}n}bebe second multiple Zagreb inde, x and forgotten index of H are; index, second multiple Zagreb inde,x and forgotten index{m,n of }H are; {m,n}
mn HM H , , ,n}} ==108 HM H{{mm,n 108mn mn PM1 H , , PM H{{mm,n = 216mn ,n}} =216 mn PM22 H H{m,n ,, PM =729 729mn m,n} = FF H = 5832 H{{m,n 2mnmn . . m,n} =583
Figure 12. 2D lattice graph of toroidal polyhex.
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3. 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 could 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 [33–35]. Some examples are Boron nanotubes, aluminosilicate/aluminugerminate, SiO2 —layered structure and titania nanotubes. Acknowledgments: This work was supported by the Dong-A University research fund in Korea. Author Contributions: Muhammad Ajmal, Waqas Nazeer, Mobeen Munir, Shin Min Kang and Young Chel Kwun contributed equally in writing this paper. Conflicts of Interest: The authors declare no conflict of interest.
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