Omega and Piv Polynomial in Dyck Graph-Like Z(8)-Unit Networks

Int. J. Nanosci. Nanotechnol., Vol. 6, No. 2, June 2010, pp. 97-103

Omega and PIv Polynomial in Dyck Graph-like Z(8)-Unit Networks

Mircea V. Diudea1, Katalin Nagy1, Monica L. Pop1, F. Gholami-Nezhaad2, A. R. Ashrafi2*

1- Faculty of Chemistry and Chemical Engineering, Babes-Bolyai University, Arany Janos Str. 11, 400084, Cluj, Romania 2- Department of Mathematics, Faculty of Science, University of Kashan, Kashan, I. R. Iran

(*) Corresponding author: [email protected] (Received: 01 Mar. 2010 and Accepted: 21 May 2010)

Abstract: Design of crystal-like lattices can be achieved by using some net operations. Hypothetical networks, thus obtained, can be characterized in their topology by various counting polynomials and topological indices derived from them. The networks herein presented are related to the Dyck graph and described in terms of

Omega polynomial and PIv polynomials.

Keywords: Dyck Graphs, Omega Polynomial, PIv Polynomial, Net Operation

1. INTRODUCTION 2. LATTICE BUILDING

Novel carbon allotropes have been discovered and The networks making - the subject of our discussion- studied for applications in nano-technology, in the were built up by the unit designed by the net last twenty years. Among the carbon structures, operation sequence Op(Q(C)), performed on the fullerenes (zero-dimensional), nanotubes (one Cube C. dimensional), graphene (two dimensional) and Recall, the Quadrupling Q (also called Chamfering) spongy carbon (three dimensional) were the most is a composite operation, which truncates the challenging materials [1, 2]. Inorganic clusters, like triangulation on the old faces of a polyhedral zeolites, also attracted the attention of scientists. object and finally deletes the original edges. The Recent articles in crystallography promoted the Q operation leaves unchanged the orientation of idea of topological description and classification of the polygonal faces. The opening operation Op crystal structures [3-8]. is achieved by adding one point of degree two on This study presents two hypothetical crystal- each of the boundary edges of the parent faces that like nano-carbon structures, with the topological become the open faces. In Op2a version (see below), description in terms of Omega and PIv counting two points are alternatively added on the boundary polynomials. edges of the parent faces. More about map/net operations, the reader can find in refs. [9-13]. This unit is the zig-zag isomer (Figure 1, left) and, together with its “armchair” isomer (Figure 1,

97 right), were designed by the sequence Op2a(Q(C)), of degree 2 were identified, thus resulting in four- and proposedThe by Diudeanetworks as representationsmaking - the subject of the of ourconnected discussion- atoms were (and builtrings); up by by this the reason unit we call celebrateddesigned Dyck by graph the [14];net operation this graph issequence built up onOp (Q(Cthis)), performeda “spiro” lattice. on the Cube C. 32 vertices of Recall,valence 3, the it has Quadrupling 48 edges, 12 octagons,Q (also calledTheChamfering) Jn net is triple is periodic a composite (Figure operation, 3) while the Id girth which6, diameter truncates 5, and the chromatic triangulation number on 2; the it old facesnet is only of a double polyhedral periodic object ((b,c,c), and in finallyFigure 2) as is non-planardeletes and the has original the minimal edges. genus The gQ = 1( operationi.e., leavesalso shown unchanged in the corner the orientationrepresentation of of the a cubic there polygonalexists an embedding faces. The of the opening graph on operation the torus). Op is domainachieved of theby latticeadding (Figure one point4, left). of degree Cycletwo counting on each on the of finitethe boundaryrepresentation edges revealed of the parent faces that become the open faces. In 12 octagons and 16 hexagons. As units of infinite These structures show large hollows, as those Op2a version (see below), two points are alternatively added on the boundary edges of lattices, the units show 12 octagons and their genus encountered in zeolites, natural alumino-silicates the parent faces. More about map/net operations,widely the reader used in can synthetic find in chemistry refs. [9-13]. as catalysts. [15] (i.e., theThis number unit is of the simple zig-zag tori isomer consisting (Figure a 1, left) and, together with its “armchair” isomer structure) is g=3. (Figure 1, right), were designed by the sequence Op2a(Q(C)), and proposed by Diudea as Thererepresentations are two ways of to the design celebrated networks Dyck using graph a [14];3. DEFINITIONS this graph is built up on 32 vertices of repeatingvalence unit: 3, (i)it hasjunction 48 edges, Jn by 12 an octagons, edge of girth the 6, diameter 5, and the chromatic number 2; it is vertices/atomsnon-planar of and degree has the 2 minimal (Figure genus 2a) and g = (ii) 1( i.e., thereIn a exists connected an embedding graph G( Vof,E the), with graph the on vertex the set identificationtorus). Cycle Id of countingsome identical on the features finite representation (Figure Vrevealed(G) and edge12 octagons set E(G ),and two 16 edges hexagons. e = uv As and f = 2,b andunits c), ofin twoinfinite close lattices, units. In the our units case, show the atoms 12 octagons xy of and G aretheir called genus codistant [15] ( i.ee co., thef if numberthey obey the of simple tori consisting a structure) is g=3.

(a) 56_Op(Q(C)) (b) 56_Op2a(Q(C))

Figure 1. DyckFigure graph-like 1: Dyck units graph-like with R[8]=12 units :with the R[8]=12:zig-zag “Z” the unit zig-zag (left) “Z” and unit the (left)armchair unit “A” (right) and the armchair unit “A” (right)

(a) There are two ways to design(b) networks using a repeating(c) unit: (i) junction Jn by an(a) edge of the vertices/atoms(b) of degree 2 (Figure 2a) and (ii) identification Id of some identical features (Figure 2,b and c), in two close units. In our case, the atoms of degree 2 were identified, thus resulting in four-connected atoms (and rings); by this reason we call this a “spiro” lattice.

Figure 2. The unit 56_Op(Q(C)) in lattice building by junction (a) and identification (b and c). FigureFigure 2:2. The unit 56_56_Op(Q(C))Op(Q(C)) in in lattice lattice building building by by junction junction (a) (a) and and identification identification (b and (b c).and c).

Figure 2. The unit 56_Op(Q(C)) in lattice building by junction (a) and identification (b and c). The Jn net is triple periodic (Figure 3) while the Id net is only double periodic ((b,c,c), 98 The Jn net is triple periodic (FigureAshrafi 3) while et al. the Id net is only double periodic ((b,c,c), in FigureThe 2) Jn as net also is showntriple periodicin the corner (Figure representation 3) while the of Id a netcubic is onlydomain double of the periodic lattice (Figure((b,c,c), 4, in Figure 2) as also shown in the corner representation of a cubic domain of the lattice (Figure 4, inleft). Figure 2) as also shown in the corner representation of a cubic domain of the lattice (Figure 4, left). left).

Figure 3. Unit (designed by Op(Q(C)) in a “Jn” (R[6],R[8]) network; the cubic domain (left) in Figureits corner 3. Unitrepresentation (designed (right). by Op (Q(C)) in a “Jn” (R[6],R[8]) network; the cubic domain (left) in Figure its corner 3. Unit representation (designed by (right). Op(Q (C)) in a “Jn” (R[6],R[8]) network; the cubic domain (left) in its corner representation (right).

Figure 4. Unit (designed by Op(Q(C)) in an “Id” (R[4],R[8]) network; the cubic domain (left) in its corner representation (right - only two pair edges of the domain are the same while the third

Figurepair is distinct,4. Unit (designedthe net being by Opdouble(Q(C)) periodic). in an “Id” (R[4],R[8]) network; the cubic domain (left) in Figure 4. Unit (designed by Op(Q(C)) in an “Id” (R[4],R[8]) network; the cubic domain (left) in its corner representation (right - only two pair edges of the domain are the same while the third itspair corner is distinct, representation the net being(right double- only twoperiodic). pair e dges of the domain are the same while the third pair is distinct, the net being double periodic).

(a) (b)

(c)

Figure 2. The unit 56_Op(Q(C)) in lattice building by junction (a) and identification (b and c).

The Jn net is triple periodic (Figure 3) while the Id net is only double periodic ((b,c,c), in Figure 2) as also shown in the corner representation of a cubic domain of the lattice (Figure 4, left).

FigureFigure 3. Unit3: Unit (designed (designed by byOp Op(Q(C))(Q(C)) in ina “Jn”a “Jn” (R[6],R[8]) (R[6],R[8]) network; network; the the cubic cubic domain domain (left) (left) in its corner representation (right). in its corner representation (right).

FigureFigure 4. 4:Unit Unit (designed (designed by by Op Op(Q(C))(Q(C)) in inan an “Id” “Id” (R[4],R[8]) (R[4],R[8]) network; network; the the cubic cubic domain domain (left) (left) in in itsits cornercorner representationrepresentation (right (right - - onlyonly twotwo pair eedgesdges of the domain areare the samesame whilewhile thethe third third pair pair is distinct, the net being isdouble distinct, periodic). the net being double periodic). relation [16]: opposite, e op f, if they are opposite edges of an inner face of G. Note that the relation co is defined = += += dvxdvy(,) (,)1 dux (,)1 duy (,) (1) in the whole graph while op is defined only in faces/ Which is reflexive, that is, e co e holds for any rings. Using the relation op the edge set of G can be edge e of G, and symmetric, i.e., if e co f then f partitioned into opposite edge strips, ops. An ops is a co e but, in general, relation co is not transitive. quasi-orthogonal cut qoc, since op is, in general, not If “co” is also transitive, thus an equivalence transitive. In co-graphs, the two strips superimpose relation, then G is called a co-graph and the to each other, then {}{}ck= sk, for any integer k. set of edges c():{();} e=∈ f E G f co e A graph G is a partial cube if it is embeddable in is called an orthogonal cut oc of G, E(G) the n-cube Qn , which is the regular graph whose being the union of disjoint orthogonal cuts: vertices are all binary strings of length n, two EG( )= c ∪ c ∪ ... ∪ c , c ∩ c =∅, i ≠ j 12 kij . strings being adjacent if they differ in exactly one Klavžar [17] has shown that relation co is a theta position [15]. The distance function in the n-cube Djoković-Winkler relation [18, 19]. is the Hamming distance. A hypercube can also be nn Two edges e and f of a plane graph G are in relation expressed as the Cartesian product: QQnn==P��i=i1=1KK22

International Journal of Nanoscience and Nanotechnology 99 For any edge e=(u,v) of a connected graph G let nuv graphs: denote the set of vertices lying closer to u than to v: ′ =⋅=⋅ nuv =∈{ wVGdwudwv()|(,)(,) <}. It follows PIv (1) e v | E ( G )|| V ( G )| (6) that nuv =∈{ wVGdwvdwu()|(,)(,)1 = +}. The sets (and subgraphs) induced by these vertices, In general graphs, this index can be written according to Ilić [40]: nuv and nvu, are called semicubes of G; the semicubes n and n are opposite semicubes and disjoint (to uv vu ' each other) [20,21]. A graph G is bipartite if and only PIv( G)= PI v(1) =∑∑ nuv,, +=⋅− n vu V E muv, if, for any edge of G, the opposite semicubes define a e= uv e= uv (7) partition of V(G): nnvVGuv+== vu (). where n , n count the non-equidistant vertices The semicubes Wab and Wba are opposite semicubes u,v v,u with respect to the endpoints of the edge e=(u,v) Let G be a connected graph and S(G)={ S,SS ,..., } be 12k while m(u,v) is the number of equidistant vertices the ops strips of G. Then the ops strips form a partition vs. u and v. However, in bipartite graphs, there are of E(G). The length of ops is taken as maximum no equidistant vertices so that the last term in (7) (among all possible ops). It depends on the size of will be missing. The value of PI (G) is thus maximal the maximum fold face/ring F /R considered, v max max in bipartite graphs, among all graphs on the same so that any result on Omega polynomial will have order; the result of (7) can be used as a criterion for this specification. checking the “bipativity” of a graph.

Denote by m(G,s) the number of ops of length s=|sk| and define the Omega polynomial as [22-30]: 4. POLYNOMINALS IN CRYSTAL-LIKE Ω(Gx,)=⋅ mGsx( ,) s (2) NETWORKS ∑ s Formulas for calculating the two above polynomials, Its first derivative (in x=1) equals the number of in the two networks here designed, are given in the edges in the graph: following tables, along with some examples. The Ω'(G ,1) =∑ mGss( , ) ⋅== e EG( ) (3) formulas were derived by inspecting the networks s included in a cubic domain (a,a,a). The indices were calculated by the Topo Cluj original program On Omega polynomial, the Cluj-Ilmenau [15] index, called Nano Studio [41] and a home-made program CI=CI(G), was defined: (supplied by A. Ilić), respectively. CI() G={ [ Ω′ (,1)] G2 −Ω [′′′ (,1) G +Ω (,1)] G } (4)

The opposite semicubes of a partial cubes represent 5. CONCLUSIONS (in general graphs) the vertex proximities of (the endpoints of) edge e=(u,v), which the PI (Ashrafi Two new hypothetical crystal-like networks were v designed, starting from the Z-isomer of a Dyck et al. [31-34]) and Cluj CJe (Diudea [35-39]) polynomials count. The PI (x) is defined as: graph representation, by using some net operations. v Their topology is described in terms of Omega and nnuv+ vu PI polynomials. Close formulas for calculating this PI() x= ∑ x (5) v v e polynomial and the derived indices were given. Relation (5) holds in any graph. The first derivative The polynomial description was proved to be useful

(in x=1) of PIv(x) gives the topological index in discriminating among structures built up from

PIv(G), which takes the maximal value in bipartite the same constructive repeat unit but following

100 Ashrafi et al.

4. 4. Polynomials Polynomials in in crystal-like crystal-like networks networks

FormulasFormulas for for calculating calculating the the two two above above polynomials, polynomials, in in the the two two networks networks here here designed,designed, are are given given in in the the following following tables tables, ,along along with with some some examples. examples. The The formulas formulas werewere derived derived by by inspecting inspecting the the networ networksks included included in in a acubic cubic domain domain ( a(,aa,,aa,).a). The The indices indices werewere calculated calculated by by the the Topo Topo Cluj Cluj original original program program called called Nano Nano Studio Studio [41] [41] and and a ahome- home- mademade program program (supplied (supplied by by A. A. Ili Iliü),ü), respectively. respectively.

TableTable 1:1. 1. Omega Omega and and PIPIv PIv v polynomials polynomialspolynomials inin in “Jn”“Jn” “Jn” (R[6],R[8]) (R[6],R[8])(R[6],R[8]) net net net(unit (unit (unit designed designed designed by by Opby Op (Op(Q(C)))Q(Q(C)))(C))) UnitUnit Formulas Formulas JnJn__aaeven 2 4 36 63a3 even ::(Jn(Jn _ _ a a , x , )x ) 3 a32 a ( a ( a 1) 1) x 4x 4 a4 36a x x 8 x8 6 xa eveneven :c 2 2 :c(Jn(Jn _ _ a aeveneven ,1) ,1) 12 12 a a (7 (7 a a 1) 1) :cc 66 32 32 :cc(Jn(Jn _ _ aeven aeven ,1) ,1) 288 288 a a 108 108 a a 36 36 a a 22 4 4 32 32 CICI( Jn( Jn _ _ aeven aeven )48(141 )48(141 a a a a 42 42 a a 3a 3 a 4a 4 a 1) 1) Jn_a 2222 Jn_aoddodd : 22 4 4 36 36 6(1)aa 6(1)aa 6(6 6(6aa 2) 2) :(_,)3(1)4Jnaxaax(_,)3(1)4Jnaxaaxoddodd ax ax 2 x2 x 6 x6 x :c 2 2 :c(Jn(Jn _ _ a aoddodd ,1) ,1) 12 12 a a (7 (7 a a 1) 1) cc 66 32 32 ::cc(Jn(Jn _ _ aodd aodd ,1) ,1) 288 288 a a 108 108 a a 60 60 a a 22 4 4 32 32 CICI( Jn( Jn _ _ aodd aodd )48(141 )48(141 a a a a 42 42 a a 3 a 3 a 4a 4 a 1) 1) 3 vv( Jn( Jn ) ) 56 56 a a3 JnJn 3 vava 2 2 56563 PIPIv (v Jn( Jn , x, )x ) e e x x 12 12 a a (7 (7 a a 1) 1) x x c c 5 5 PIPIv v( Jn( Jn ,1) ,1) e e v v 672 672 a a (7 (7 a a 1) 1)

TableTable 2. 2. Examples Examples to to the theTable formulas formulas 2: Examples in in Table Table to 1. the1. formulas in Table 1. aaeveneven OmegaOmega (R (Rmaxmax[8])[8]) v v e e CICI PI PIv v 22 12x 12x4+32x4+32x6+8x6+8x4848 448448 624 624 369600 369600 279552 279552 44 144x 144x4+256x4+256x6+8x6+8x384384 35843584 5184 5184 25682688 25682688 18579456 18579456 66 540x 540x4+864x4+864x6+8x6+8x12961296 1209612096 17712 17712 300238272 300238272 214244352 214244352 aaoddodd 33 54x 54x4+108x4+108x6+2x6+2x144144+6x+6x168168 1512 1512 2160 2160 4450032 4450032 3265920 3265920 55 300x 300x4+500x4+500x6+2x6+2x720720+6x+6x760760 7000 7000 10200 10200 99514800 99514800 71400000 71400000 77 882x 882x4+1372x4+1372x6+2x6+2x20162016+6x+6x20722072 19208 19208 28224 28224 762643056 762643056 542126592 542126592

TableTableTable 3. 3. 3:Omega Omega Omega polynomial polynomial polynomial in in inId, Id, Id, spiro spiro spiro (R[4],R[8]) (R[4],R[8]) (R[4],R[8]) net net (unitnet (unit (unit designed designed designed by by Op Opby(Q( Op(Q(C)))Q(C)))(C)))

22 12 12aa 2 2 16 16 aa aa 16 162 2 ::(Idspiroxax(Idspiroxax , , ,) ,) 2 2 2 2ax ax ax ax 3 :c(Id , spiro ,1) 72 a 32 :cc(Id , spiro ,1) 8 a (32 a 100 a 9) CI( Id , spiro ) 32 a42 (162 a 8 a 25) 22 vIDspiro( , ) a (104 48( a 2)) 8 a (6 a 1)

v 3 8aa2 (6 1) PIv ( Id , spiro , x ) e x 72 a x

c 5 PIv (, Idspiro ,1) ev 576(6 a a 1)

Table 4. Examples of the formula in Table 3. International Journal of Nanoscience and Nanotechnology 101 a Omega (Rmax[8]) v e CI PIv 3 18x36+18x48+3x144 1368 1944 3652128 2659392 4 32x48+32x64+4x256 3200 4608 20766720 14745600 5 50x60+50x80+5x400 6200 9000 79700000 55800000

5. Conclusions

Two new hypothetical crystal-like networks were designed, starting from the Z- isomer of a Dyck graph representation, by using some net operations. Their topology is described in terms of Omega and PIv polynomials. Close formulas for calculating this polynomial and the derived indices were given. The polynomial description was proved to be useful in discriminating among structures built up from the same constructive repeat unit but following different schemes of assembling, e.g. joining “Jn” or identifying “Id” ones. Formulas for the structure parameters as: the number of atoms/vertices and bonds/edges were also derived and examples of calculations were given.

Acknowledgements: The work was supported in part by the Romanian Grant CNCSIS PN-II IDEI 506/2007.

References 1. M. V. Diudea, Ed., Nanostructures, novel architecture, NOVA, 2005. 3 :c(Id , spiro ,1) 72 a 32 :cc(Id , spiro ,1) 8 a (32 a 100 a 9) 42 CIIdspiro( , ) 32 a42 (162 a 8 a 25) 22 vIDspiro( , ) a (104 48( a 2)) 8 a (6 a 1) 2 v 3 8aa2 (6 1) PIv ( Id , spiro , x ) e x 72 a x

PIc(, Idspiro ,1) ev 576(6 a5 a 1) PIv (, Idspiro ,1) ev 576(6 a a 1)

Table 4. Examples ofTable the formula 4: Examples in Table of 3.the formula in Table 3.

a Omega (Rmax[8]) v e CI PIv 36 48 144 3 18x36+18x48+3x144 1368 1944 3652128 2659392 48 64 256 4 32x48+32x64+4x256 3200 4608 20766720 14745600 60 80 400 5 50x60+50x80+5x400 6200 9000 79700000 55800000

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