Abstract: A map M is a combinatorial representation of a graph embedded on a surface and creates a tessellation on that surface. The map operations are topological transformations of a given map. A computer program, CVNET, was made for generating maps or coverings representing fullerenes/ hyper structures, by map operations. In this paper are presented algorithms for map operations, which analyze the original graph and then generate the new ones. Some examples of generated objects by map operations are illustrated at the end

Liliana Fodor

Monica Fodor

Monica Stefu

T Rusu

CiteSeerX

 575 
Bulletin UASVM, Horticulture 65(2)/2008 
pISSN 1843-5254; eISSN 1843-5394 
 
 
CVNET SOFTWARE PROGRAM AND GENERATED STRUCTURES 
 
Monica FODOR, Monica STEFU, Liliana FODOR, T. RUSU  
 
Technical University Cluj-Napoca, 103-105 B-dul Muncii, Cluj-Napoca  
mstefu02@yahoo.com 
 
Keywords: algorithms for map operations, surface covering, nanostructures 
 
Abstract: A map M is a combinatorial representation of a graph embedded on a surface and creates a 
tessellation on that surface. The map operations are topological transformations of a given map. A computer 
program, CVNET, was made for generating maps or coverings representing fullerenes/ hyper structures, by map 
operations. In this paper are presented algorithms for map operations, which analyze the original graph nd then 
generate the new ones. Some examples of generated objects by map operations are illustrated at the end.
 
INTRODUCTION  
 
The Fullerenes are molecules composed entirely of carbon of valence 3, in the form of a 
hollow sphere, ellipsoid, tube or their derivations and combinationsi, with different coverings. 
A classical fullerene called also buckyball has a pseudo-spherical shape made up entirely of 
12 pentagons and various numbers of hexagons (Figure 1 a), as results from the Euler’s 
formulas (see below). Cylindrical fullerenes called carbon nanotubes are constructed mostly 
from hexagonsii, but they often contain defects as non hexagonal polygons (Figure 1 b). 
C60 classical fullerene (a) Armchair Polyhex Nanotubes (b) 
  
Figure 1 
A buckyball fullerene (a) and a nanotube (b) 
A graph is said to be mbedded in a surface S when it is drawn on S so that no two edges 
intersectiii . A graph is planar if it can be embedded in a plane. A map M is a combinatorial 
representation of a graph embedded on a surfaceiv nd creates a tessellation on that surface. 
The map operations are geometrical and topological transformations of a given map. Like the 
graph notations, we denote in a map: v – the number of vertices, e – the number of edges, f – 
the number of faces and d – the vertex degree. There are some basic relations in a mapv: 
∑ ∑ == esfdv sd 2     
where vd is the number of vertices of degree d and fs the number of faces with s edges, 
respectively. The two relations are joined in the Euler’s formulavi: 
  )1(2)( gMfev −==+− χ    
where χ is the Euler’s characteristic and g the genus3 of the graph (i. e.,  g = 0 for a planar or 
spherical graph and 1 for a toroidal graph). Positive/negative χ values indicate positive/ 
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negative curvature of a map or lattice. This formula is useful for checking the consistency of an 
assumed structure. For example, the C60 fullerene has vertices of degree 3 v3=60, edges e=90 
and faces of size 5 and 6 f5=12, f6=20. For molecular modeling, the fullerenes are represent d 
by molecular graphs embedded in surfaces or correspnding maps. Thus the surfaces are covered with 
diverse patterns by the polygons or faces of the maps, obtaining hyper structures. As 
theoretical models, fullerenes with faces of various sizes and coverings have been 
proposedvii,viii . Some of them are obtained by map operations or by joining fullerenes 
fragmentsix. 
A computer program, CVNETx, with options for map operations was made. A few map 
operations implemented in the CVNET program are presented in the following section. 
Section Algorithms used for Map Operations describes and presents some of the algorithms 
used by the CVNET program.  
 
MATERIAL AND METHODS  
 
The Map Operations can be classified in simple, composite, generalized and others 
operations2,7. Dualization Du of a map is built as follows: locate a point in the centre of each 
face and join two such points if their corresponding faces share a common edge. The new 
map is called the dual Du(M). The following relations exist between the elements of the 
initial and resulting map4: 
Du(M):  0fv = ;  0ee= ;  0vf =  
where subscript index “0” mark the corresponding parameters in the parent map.  
Cube Dual(Cube) = Octahedron Cubeoctahedron =  Me(Cube) 
 
Figure 2  
Results of the Dual and Medial operations on the Cube 
 
Medial Mexi puts the new vertices as the midpoints of the original edges and join two 
vertices if and only if the original edges span an angle on a face in the original map. The 
resulting parameters are:    
Me(M):  0ev = ;  02ee= ;  00 vff +=    
Figure 2 illustrates Dual and Medial operations on the Cube. Others simple operations 
are Truncation, Stellation, P4 Capping, and P5.  
The composite operationsxii can be obtained by sequences of some simple operations. 
The vertex multiplication factor m = v/v0 for a composed or generalized operation that can be 
denoted by (a,b)xiii , applied on a 3-valent map can be counted by the Goldberg’s relationxiv: 
0;);( 22 >+≥++= babababam    
The Quadrupling Q11,xv and the Capra Caxvi operations on the Cube are illustrated in the 
Figure 3. The multiplication factors for trivalent maps are m(2,0) = 4 and m(2,1) =7, 
respectively. Capra is a chiral operation (has two variants) and generates chiral objects. 
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Q(Cube)  (a) Ca(Cube)  (b) 
  
Figure 3 
 Quadrupling (a) and Capra (b) operations on the Cube 
Other chiral operations are some g neralized operations13 noted by (a,b), where a ≠ b and 
a≠ 0. The (3,1) generalized operation (Figure 4) has the multiplication factor m=9+3+1=13.  
(3,1)1 (Cube) (3,1)2 (Cube) 
 
 
Figure 4 
 (3,1) operation has two variants, resulting in chiral objects. 
The (a,b) operations, where a = b or a = 0, generates achiral objects. Some of the 
generalized operations can be obtained by successions of composite operations.   
 
Supra-Coverings7 
A perfect Clar structurexvii,xviii  PC is a disjoint set of faces, built up on all vertic s in M, 
whose boundaries form a regular 2-valent spanning subgraph. A PC structure is associated 
with a Fries structurexix, which is a spanning subgraph of the initial structure from which the 
edges from PC structure have been eliminated. A trivalent polyhedral graph, like that of 
fullerenes, has a PC structure if and only if it has a Fries structure11. Leapfrog Le is the only 
operation that provides PC results. Figure 5 presents Perfect Clar and Fries structures of C180. 
Le(C60) = Le(Le(C20)) = (3,0) (C20) 
= C180, PC structure (a) 
Le(C60) =C180 ,  Fries structure 
(b) 
Q(C60) = Q(Le(C20)) = C240,  
Pcor flowers (c) 
   
Figure 5 
Perfect Clar PC (a) and Fries structures of C180 (b); Perfect Coranulenic PCor structure of C240 (c) 
By extension, a coranulenic system8 was considered (Figure 5 c). A Perfect Coranulenic 
structure PCor is a disjoint set of (supra) faces covering all vertices in the molecular graph. 
The operation sequence Le(Q(M)) = Q(Le(M)), that is equivalent to the (2,2) generalized 
operation, provide a PCor structure. The PCor transform superimposes over PC, thus, any 
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PCor is necessarily a PC. The PC and PCor structure is expected to contribute to the stability 
of the whole molecule. Others operations, like Capra, Disjoint Azulenic operation and 
Rotated Disjoint Azulenic operation generates others kinds of “flowers”.  
Algorithms used for Map Operations 
CageVersatile .Net CVNET software program, written in C#, under .NET Framework, 
is a program that has options for map operations and it generates closed or open lattices 
covering nanostructures. The input and output files are in HyperChem – .hin format and 
contain the representation of labelled graphsxx.  
First, the program finds the rings or the faces of the input graph and then it realizes the 
map operations on every face. To find the faces in a map, the program uses a recursive 
algorithm to detect all the cycles of the graph with length less or equal to a input parameter. 
In this respect, the program uses the table of links between vertices from the input file. Then 
the program selects the cycles as f ces, in ascending order of their size, under the condition 
that any edge shares maximum 2 faces. Next, the algorithm of map operations works in the
following steps, for every face or cycle: 
1. Put the vertices of the resulting map at their coordinates, calculated function of the 
coordinates of the vertices from the initial map, for every old face. 
2. Link the vertices located inside of an old face. 
3. Link the vertices located in different old faces. In case of chiral resulting objects, the old 
faces are visited by adjacency, so that the proper links are accomplished (Figure 6). ..
. .
 
.
. .
.
 
.
. .
.
 
..
. .
 
Step 1.  
Put the new vertices  
Step 2.  
Link the inner vertices 
Step 3. Link the above linked vertices with the  
external or border vertices  
The algorithm steps of the Capra operation on a squre face                           Figure 6  
We detail below the general algorithm that creates th  new map on the basis of the old 
faces, used at composite, generalised and others complex operations. The variable c is a three-
dimensional array, containing the faces of the graph. In the array c, c[i,0,0] is the number of 
faces contained in the vector c[i], which begins at the vertex i; nv = c[i,j,0] is the number of 
vertices of the face c[i,j], while c[i,j,1] ... c[i,j,nv] is the list of vertices of the face c[i,j]. The 
variable n stores the number of vertices in the input map. Similary like c, m stores the list of 
edges of the initial molecular graph and some additional new vertices, used within operations;  
m[i,j] represents an edge with the end-points in m[i,j,1] and m[i,j2]. The variable cip is an 
array that will store the resulting graph, with vertices, their coordinates and the links between 
vertices. 
  for i := 1 to n    
    for j := 1 to c[i,0,0]  
      nv := c[i,j,0]; 
      coord_center(c[i,j], cc);  // the array cc will store the 3D coordinates (x,y,z) of the center of the face  
      for k := 1 to nv  
   v_next := (k mod nv) +1;  // the vertex following vertex no. k in the face  
      //add the new vertices that are inside the fac and between the center and the vertices c[i,j,k], c[i,j,v_next] 
add_vertices_inside(c[i,j,k], c[i,j,v_next], cc, cip);  
      endfor k 
 579 
      for k := 1 to nv  
 v_next := (k mod nv) +1;   // link the vertices tha  are between the vertices c[i,j,k], c[i,j,v_next] 
 link_vertices_inside(c[i,j,k], c[i,j,v_next], cip);  
      endfor k 
    endfor j 
  endfor i 
  for i := 1 to n 
    for j := 1 to m[i,0] 
      add_vertices(m,i,j,cip);  // adds the rest of he border vertices that are close to the edge m[i,j]  
    endfor j                          // and link them if possible 
  endfor i     //in case of chiral operation, visit the faces by adjacency and link the border vertices,  
 // otherwise the algorithm stops here.  
  nc := 0; c[i,j,nv+2] :=0;   // nc numbers the visited faces;  all the faces are marked with “0”, as non -visited. 
  do while nc < nci // do while exists a face not visited; nci is the number of faces in the original graph. 
    for i := 1 to n  // take all the faces in the array c 
      for j := 1 to c[i,0] 
        nv := c[i,j,0]; 
        if (nc = 0) or ((c[i,j,nv+2] =0) and (AlreadyVisited(c,i,j,m)) then 
          // if it is the first face or the current unvisited face shares an edge with an already visited face 
          nc := nc + 1; // visit the face i.e. link the border vertices from different faces 
          c[i,j,nv+2] := 10;  // mark the face as vi ited 
// the border vertices from the adjacent visited face can link in 2 ways 
          link := verify_linking_mode(c,i,j,m);  
          for k := 1 to nv  
add_border _links(link,c,i,j,k,m,cip); // add links to the border vertices from the current face 
         endfor k 
        endif 
      endfor j 
    endfor i 
  loop 
  WriteF(cip);    // writes the array cip into the output file. 
For map operations that generates supra-coverings “flowers”, the program has options 
to generate the whole map or/and only the “flowers” structure and eventually also the 
complementary structure. After generating the whole structure, it is optimized to get a 
rounded shape, and then coordinates of the whole structure can be transferred to the structure 
of “flowers” to get the same shape. For any object, it can be checked if it has a Perfect Clar 
PC or Perfect Coranulenic PCor structure. 
 
RESULTS AND DISCUSIONS 
 
A large variety of objects of different genuses and coverings can be obtained by 
successions of map operations. In the following figures are presented some examples. 
 (2,2) operation on a structure starting from tetrahedrons (2,2) (Op(Ca(C20)))   g=6 
  
Disjoint Corannulenic Structures (PCor) obtained by faces opening (Op), starting from Dodecahedron/C20 Figure 7 
 580 
(2,2) (Le(Op(Ca(Tetrahedron)))) S2 (S2(Op(Ca(Cube)))) S2 (S2 (S2(C20))) 
   
Figure 8  
Other objects obtained by map operations 
 
The program works on any faces sizes and vertices degree. There is an option to 
consider all the cycles that have the length less or equal the input parameter, not only the 
faces. Full objects can be generated in this way.  
 
CONCLUSIONS 
 
Various molecular graphs/maps/hyper structures, with diverse tiling and genuses, were 
generated by CVNET computer program for map operations. They were optimised and were 
further topological analysed from stability point of view.   
 
REFERENCES 
 
                                                
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CVNET SOFTWARE PROGRAM AND GENERATED STRUCTURES

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CVNET SOFTWARE PROGRAM AND GENERATED STRUCTURES

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