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 periodica composite (Figure operation, 3) while the Id girth which6, diameter truncates 5, and the chromatictriangulation number on 2;the it old facesnet is onlyof a doublepolyhedral periodic object ((b,c,c), and in finallyFigure 2) as is non-planardeletes andthe hasoriginal 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 isof thesimple zig-zag tori isomerconsisting (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 ofto thedesign celebrated networks Dyck using graph a [14];3. DEFINITIONSthis graph is built up on 32 vertices of repeatingvalence unit: 3, (i)it hasjunction 48 edges, Jn by 12 an octagons, edge of girththe 6, diameter 5, and the chromatic number 2; it is vertices/atomsnon-planar of and degree has the 2 minimal(Figure genus2a) and g =(ii) 1( i.e., thereIn a existsconnected 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. (c) (c) (c) 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 Figureits 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.