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Light Paths in 4Connected Graphs in the Plane and Other Surfaces
 J. Graph Theory
, 1998
"... Several results concerning existence of kpaths, for which the sum ..."
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Several results concerning existence of kpaths, for which the sum
Encoding fullerenes and geodesic domes
 SIAM. J. Discrete Math
, 2004
"... Abstract. Coxeter’s classification of the highly symmetric geodesic domes (and, by duality, the highly symmetric fullerenes) is extended to a classification scheme for all geodesic domes and fullerenes. Each geodesic dome is characterized by its signature: a plane graph on twelve vertices with label ..."
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Abstract. Coxeter’s classification of the highly symmetric geodesic domes (and, by duality, the highly symmetric fullerenes) is extended to a classification scheme for all geodesic domes and fullerenes. Each geodesic dome is characterized by its signature: a plane graph on twelve vertices with labeled angles and edges. In the case of the Coxeter geodesic domes, the plane graph is the icosahedron, all angles are labeled one, and all edges are labeled by the same pair of integers (p, q). Edges with these “Coxeter coordinates ” correspond to straight line segments joining two vertices of Λ, the regular triangular tessellation of the plane, and the faces of the icosahedron are filled in with equilateral triangles from Λ whose sides have coordinates (p, q). We describe the construction of the signature for any geodesic dome. In turn, we describe how each geodesic dome may be reconstructed from its signature: the angle and edge labels around each face of the signature identify that face with a polygonal region of Λ and, when the faces are filled by the corresponding regions, the geodesic dome is reconstituted. The signature of a fullerene is the signature of its dual. For each fullerene, the separation of its pentagons, the numbers of its vertices, faces, and edges, and its symmetry structure are easily computed directly from its signature. Also, it is easy to identify nanotubes by their signatures.
Zigzag structure of simple twofaced polyhedra
 Combinatorics, Probability & Computing 14
, 2005
"... A zigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbors on opposite edges. A graph without a railroad is called tight. ..."
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A zigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbors on opposite edges. A graph without a railroad is called tight. We consider the zigzag and railroad structures of general 3valent plane graph and, especially, of simple twofaced polyhedra, i.e., 3valent 3polytopes with only agonal and bgonal faces, where 3 ≤ a < b ≤ 6; the main cases are (a,b) = (3,6), (4,6) and (5,6) (the fullerenes). We completely describe the zigzag structure for the case (a,b)=(3,6). For the case (a,b)=(4,6) we describe symmetry groups, classify all tight graphs with simple zigzags and give the upper bound 9 for the number of zigzags in general tight graphs. For the remaining case (a,b)=(5,6) we give a construction realizing a prescribed zigzag structure.
Fullerenes as tilings of surfaces
 J. Chem. Inf. Comput. Sci
"... If a fullerene is defined as a finite trivalent graph made up solely of pentagons and hexagons, embedding in only four surfaces is possible: the sphere, torus, Klein bottle, and projective (elliptic) plane. The usual spherical fullerenes have 12 pentagons; elliptic fullerenes, 6; and toroidal and Kl ..."
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If a fullerene is defined as a finite trivalent graph made up solely of pentagons and hexagons, embedding in only four surfaces is possible: the sphere, torus, Klein bottle, and projective (elliptic) plane. The usual spherical fullerenes have 12 pentagons; elliptic fullerenes, 6; and toroidal and Kleinbottle fullerenes, none. Kleinbottle and elliptic fullerenes are the antipodal quotients of centrosymmetric toroidal and spherical fullerenes, respectively. Extensions to infinite systems (plane fullerenes, cylindrical fullerenes, and space fullerenes) are indicated. Eigenvalue spectra of all four classes of finite fullerenes, are reviewed. Leapfrog fullerenes have equal numbers of positive and negative eigenvalues, with 0, 0, 2, or 4 eigenvalues zero for spherical, elliptic, Kleinbottle, and toroidal cases, respectively.
2004), DoubleLink Expandohedra: A Mechanical Model for Expansion of a Virus, submitted to
 Proceedings of the Royal Society: Mathematical, Physical & Engineering Sciences
"... Doublelink expandohedra are introduced: each is constructed from a parent polyhedron by replacing all faces with rigid plates, adjacent plates being connected by a pair of spherically jointed bars. Powerful symmetry techniques are developed for mobility analysis of general doublelink expandohedra ..."
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Doublelink expandohedra are introduced: each is constructed from a parent polyhedron by replacing all faces with rigid plates, adjacent plates being connected by a pair of spherically jointed bars. Powerful symmetry techniques are developed for mobility analysis of general doublelink expandohedra, and when combined with numerical calculation and physical model building, demonstrate the existence of generic finite breathing expansion motions in many cases. For icosahedrally symmetric trivalent parents with pentagonal and hexagonal faces only (fullerene polyhedra), the derived expandohedra provide a mechanical model for the experimentally observed swelling of viruses such as cowpea chlorotic mottle virus (CCMV). A fully symmetric swelling motion (a finite mechanism) is found for systems based on icosahedral fullerene polyhedra with odd triangulation number, T ≤ 31, and is conjectured to exist for all odd triangulation numbers.
4valent plane graphs with 2, 3 and 4gonal faces
 SATELLITE CONFERENCE (OF CONGRESS OF ISM IN BEIJING) ON ALGEBRA AND COMBINATORICS
, 2003
"... Call ihedrite any 4valent nvertex plane graph, whose faces are 2, 3 and 4gons only and p2+p3 = i. The edges of an ihedrite, as of any Eulerian plane graph, are partitioned by its central circuits, i.e. those, which are obtained by starting with an edge and continuing at each vertex by the edg ..."
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Call ihedrite any 4valent nvertex plane graph, whose faces are 2, 3 and 4gons only and p2+p3 = i. The edges of an ihedrite, as of any Eulerian plane graph, are partitioned by its central circuits, i.e. those, which are obtained by starting with an edge and continuing at each vertex by the edge opposite the entering one. So, any ihedrite is a projection of an alternating link, whose components correspond to its central circuits. Call an ihedrite irreducible, if it has no railroad, i.e. a circuit of 4gonal faces, in which every 4gon is adjacent to two of its neighbors on opposite edges. We present the list of all ihedrites with at most 15 vertices. Examples of other results: (i) All ihedrites, which are not 3connected, are identified. (ii) Any irreducible ihedrite has at most i − 2 central circuits. (iii) All ihedrites without selfintersecting central circuits are listed. (iv) All symmetry group of ihedrites are listed.
Fullerenes and Coordination Polyhedra versus HalfCubes Embeddings
, 1997
"... A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking fo ..."
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A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes F n for n ! 60 and of all preferable fullerenes C n for n ! 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onionlike metallic clusters and geodesic domes. Quasiembeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells. Contents 1 Introduction and Basic Properties 2 1...
Version of Zones and Zigzag Structure in Icosahedral Fullerenes and Icosadeltahedra
, 2002
"... A circuit of faces in a polyhedron is called a zone if each face is attached to its two neighbors by opposite edges. (For oddsized faces, each edge has a left and a right opposite partner.) Zones are called alternating if, when odd faces (if any) are encountered, left and right opposite edges are c ..."
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A circuit of faces in a polyhedron is called a zone if each face is attached to its two neighbors by opposite edges. (For oddsized faces, each edge has a left and a right opposite partner.) Zones are called alternating if, when odd faces (if any) are encountered, left and right opposite edges are chosen alternately. Zigzag (Petrie) circuits in cubic () trivalent) polyhedra correspond to alternating zones in their deltahedral duals. With these definitions, a full analysis of the zone and zigzag structure is made for icosahedral centrosymmetric fullerenes and their duals. The zone structure provides hypercube embeddings of these classes of polyhedra which preserve all graph distances (subject to a scale factor of 2) up to a limit that depends on the vertex count. These embeddings may have applications in nomenclature, atom/vertex numbering schemes, and in calculation of distance invariants for this subclass of highly symmetric fullerenes and their deltahedral duals. 1.
Faceregular 3valent twofaced spheres and tori
"... Call twofaced map and, speci¯cally, (p; q)map a 3valent map (on sphere or torus) with only p and qgonal faces (at least one each), for given integers 3 · p < q; so, 3 · p · 5. Twofaced maps (especially, (5; 6)polyhedra, called fullerenes) are prominent molecular models in Chemistry. We say ..."
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Call twofaced map and, speci¯cally, (p; q)map a 3valent map (on sphere or torus) with only p and qgonal faces (at least one each), for given integers 3 · p < q; so, 3 · p · 5. Twofaced maps (especially, (5; 6)polyhedra, called fullerenes) are prominent molecular models in Chemistry. We say that a (p; q)map is pRi if any pgon has the same number i of pgonal neighbors; it is qRj if each qgon has the same number j of qgonal neighbors. Call a (p; q)map strictly faceregular if it is both, pRi and qRj, for some i,j; call it weakly faceregular, if it is only pRi or qRj. All strictly faceregular (p; q)polyhedra are ([BrDe99], [De02]) Prismm, Barrelm (m ¸ 3) and 55 sporadic polyhedra. By Barrelm we denote 4mvertex (5; m)polyhedron, consisting of two mgons separated by two mrings of 5gons. All 23 parametersets (p; q; i; j) for strictly faceregular (p; q)tori are found ([De02]); the number of minimal tori is one for 7 of them and an in¯nity for 16 others. Here we address the characterization of all weakly faceregular (p; q)maps on sphere or torus. Examples of obtained results are: 1. Any (3; q)map, which is 3R0, has 4 · q · 12. It is strongly faceregular for q = 4; 5 (Prism3 and Barrel3 only) and q = 11; 12 (only tori, unique for q = 12). All such weakly faceregular maps are in¯nities of polyhedra and tori for each 7 · q · 10 and (characterized) in¯nity of (3; 6)polyhedra. 2. Weakly faceregular (5; q)polyhedra 5Rj exist for j = 3; 6 · q · 10, and j = 2; q ¸ 8. 3. The following general conjecture: the number of (p; q)polyhedra, which are qRj, is in¯nite if and only the corresponding torus exist. 4. If a (p; q)polyhedron is qRj, then j · 5. The number of (5; q)polyhedra is