Fantasmagoric Fulleroids Revisited
Mircea V. DIUDEA
Faculty of Chemistry and Chemical Engineering,
“BabesBolyai” University, 400028, Cluj, Romania
Abstract
New strange fulleroids are built up by using the three classical composite map operations: tripling (leapfrog Le), quadrupling (chamfering Q) and septupling (capra Ca) on the starting structure dodecahedron. These transforms belong to the icosahedral symmetry group and show interesting mathematical and (possible) physicochemical properties. Topological symmetry data of the presented cages are calculated by means of layer matrices derived on the associated graphs and their medial and dual transforms.
Keywords
Fulleroids, Composite operations on maps, Graphs
Introduction
A map M is a combinatorial representation of a closed surface [1]. Several transformations (i.e., operations) on maps are known and used for various purposes.
Let us denote in a map: v  number of vertices, e  number of edges, f  number of faces and d  vertex degree. A subscript “0” will mark the corresponding parameters in the parent map.
Recall some basic relations in a map [2]:
ådn_{d} = 2e (1)
åsf_{s} = 2e (2)
where v_{d} and f_{s} are the number of vertices of degree d and number of sgonal faces, respectively. The two relations are joined in the famous Euler [3] formula:
ν – e + f = χ(M) = 2(1g) (3)
with c being the Euler characteristic and g the genus [4] of a graph (i.e., the number of handles attached to the sphere to make it homeomorphic to the surface on which the given graph is embedded; g = 0 for a planar graph and 1 for a toroidal graph). Positive/negative c values indicate positive/negative curvature of a lattice. This formula is useful for checking the consistency of an assumed structure.
It is assumed that simple map operations, like dualization, stellation, truncation, medial, etc. are wellknown; the reader is referred to some already published papers [5].
Composite Operations on Maps
Leapfrog Le is a composite operation [6][10] which can be written as:
Le(M) = Du(P_{3}(M)) = Tr(Du(M)) (4)
A sequence of stellation (i.e., P_{3})  dualization rotates the parent sgonal faces by p/s. Leapfrog operation is illustrated, for a tetragonal face, in Figure 1.
A bounding polygon, of size 2d_{0}, is formed around each original vertex. In the most frequent cases of 4 and 3valent maps, the bounding polygon is an octagon and a hexagon, respectively.
Figure 1. Le operation on a tetragonal face of a trivalent net
In trivalent maps, Le(M) is the tripling operation. The complete transformed parameters are:
Le(M): n = s_{0}f_{0} = d_{0}n_{0}; e = 3e_{0}; f = n_{0} + f_{0} (5)
Quadrupling Q (i.e., chamfering [11]) is another composite operation, achieved by the sequence [12]:
Q(M) = E_(Tr_{P3}(P_{3}(M))) (6)
Figure 2. Q operation on a tetragonal face of a trivalent net
The complete transformed parameters are:
Q(M): n = (d_{0} + 1)n_{0}; e = 4e_{0}; f = f_{0} + e_{0} (7)
Q operation leaves unchanged the initial orientation of the polygonal faces. Note that, the quadrupling transform of a 4valent map is not a regular graph anymore (because of mixing the new trivalent vertices with the parent 4valent ones). Only Q(M) of a 3valent map is a 3regular graph.
Q insulates the parent faces always by hexagons.
capra Ca  the goat, is the romanian corresponding of the leapfrog English children game. It is a composite operation, 12 necessarily coming from the Goldberg’s 11 multiplying factor m:
M = (a^{2} + ab + b^{2}); a ³ b; a + b > 0 (8)
that predicts m (in a 3valent map) as follows: Le: (1, 1); m = 3; Q: (2, 0); m = 4; Ca: (2, 1); m = 7.
The transformation can be written as 12:
Ca(M) = Tr_{P5}(P_{5}(M)) (9)
with Tr_{P}_{5 }meaning the truncation of each vertex where P_{5} capping faces of a parent face are incident.
Ca insulates any face of M by its own hexagons, which are not shared with any old face (in contrast to Le or Q).
The operation is illustrated in Figure 3. Note that Ca is S_{1} operation which is different to S_{2} twin operation [13].
Figure 3. Ca operation on a tetragonal face of a trivalent lattice
Open Operation
The above operations Ω(M) can continue by an E_{n} homeomorphic transformation E_{n}(Ω(M)) of the boundary edges of the parentlike faces thus resulting open maps with all polygons of the same size. In case of E_{1} operation the size is: 9 (Le); 8 (Q) and 7 (Ca). This opening transformation is illustrated in Figures 4 to 6 in case of the dodecahedron D.
The opening faces by E_{1} operation are “zigzag” tubule junctions. They can be further decorated by halves of the dodecahedron, thus entering pentagonal faces. Figures 7 to 9 illustrate the resulting objects, both in 3D and in 2D, as Schlegel projection [14].
E(Le(D))); v = 120 
E(Le(D))); Schlegel projection 
Figure 4. Open of the Le(D)
E(Q(D))); v = 140 
E(Q(D))); Schlegel projection 
Figure 5. Open of the Q(D)
E(Ca(D))); v = 200 
E(Ca(D))); Schlegel projection 
Figure 6. Open of the Ca(D)
Dh(E(Le(D))); v = 180 

Figure 7. The cage derived on Le(D) by opening the original faces and decorating by halves of D
Dh(E(Q(D))); v = 200 

Figure 8. The cage derived on Q(D) by opening the original faces and decorating by halves of D
Dh(E(Ca(D))); v = 260 

Figure 9. The cage derived on Ca(D) by opening the original faces and decorating by halves of D
Discussion
Given a graph G with S = S(G) being the set of subgraphs of a given size and a group of automorphisms Aut(G(S)), two subgraphs S_{i}, S_{j} of G are called equivalent if there is a group element aut(N_{Si}) Í Aut(G(S)) such that S_{j}_{ }aut(N_{Si}) S_{i }(_{ }i.e., an automorphic permutation that transforms two subgraphs to each other).
The set of all subgraphs S_{j} obeying the above equivalence relation is called the orbit (i.e., class of equivalence) of the subgraph S_{i}, S_{NSi}. The orbits represent disjoint automorphic partitions: S = ÈS_{NSi}. We limit here to find the equivalence classes for the vertices, edges and faces of these cages. As an equivalence criterion we used the index of centrality C(LM) defined on layer matrices LM [15][17].
The vertex orbits are found by applying the above index on the layer matrix of valences L^{1}W of the parent graph. The edge orbits are calculable with the same index but applied on the corresponding medial graph.
The face orbits are obtained from the dual of the parent graph. Each vertex in the dual has the valence equal to the parent face size. In this way one can select the orbits for each face type. Topological symmetry data are listed in Table 1.
Table 1. Equivalence classes in phantasmagoric fulleroids
G 
Vertices 
Edges 
Faces 
Le 
180; 3x{60} 
270; {30};2x{60};{120} 
92; {12},[5]; {60},[5]; {20},[9] 
Q 
200; {20}; 3x{60} 
300; 3x{60};{120} 
102; {12},[5]; {60},[5]; {30},[8] 
Ca 
260; {20}; 4x{60} 
390; {30};6x{60} 
132; {12},[5]; {60},[5]; {60},[7] 
The two first operations Le and Q applied on the dodecahedron lead to achiral icosahedral cages, belonging to the I_{h} point group symmetry. However, the third transformation Ca preserves the icosahedral rotational symmetry but destroys all reflection operations thereby leading to the chiral point group I. More about the symmetry preserving by the above map operations the reader can find in ref [18].
According to the Clar theory, any polyhedral map may be looked for a perfect Clar PC structure [19], [20] which is a disjoint set of faces (built up on all vertices in M) whose boundaries form a 2factor. Recall that a kfactor is a regular kvalent spanning subgraph. A PCS is associated with a Fries structure which is a Kekulé valence structure having the maximum possible number of benzenoid faces. Such structures represent total resonant sextet TRS benzenoid molecules and it is expected to be extremely stable, in the valence bond VB theory [21], [22]. Leapfrog Le is the only operation that provides PC transforms.
By extension, a corannulenic structure can be imagined 12 within the VB theory, either as (i) intersected, (ii) joint or (iii) disjoint units. While the case (i) is trivial, joint corannulenic JCor structures appear by Ca operation. A disjoint corannulenic DCor structure is a disjoint set of flowers, covering all the vertices in the molecular graph; it is a connected equivalent of the Fries structure. Only the (2,2) generalized [23] operation and/or (Le, Q) sequence provide a DCor covering [24]. Such a transform is necessarily associated with a PC structure, but the reciprocal is not always true. Observe the structure Dh(Le(D)), Figure 7, which is a disjoint structure of dodecahedron halves. In a next paper, Kekulé valence structure counting of Dh(E(Le(D))) will be discussed in detail into a future paper.
Our CageVersatile program enables performing the map operations on surfaces of any genus and vertex degree lattice.
Conclusions
Map operations enable construction of various cages starting from the Platonic solids. Moreover, they can be used as background for rationalizing the “in silico” building up of some previously published cages, like the phantasmagoric fulleroid of Fowler, Dh(E(Ca(D))). Topological symmetry data of the presented cages are available by performing some layer matrices on the associated graphs and their medial and dual transforms. The structure Dh(E(Le(D))), having disjoint dodecahedron halves, is a possible candidate to real molecular structure since the dodecahedron was already isolated as C_{20} molecule.
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