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Fullerene Graphs With More Negative Than Positive Eigenvalues; The Exceptions That Prove The Rule of Electron De
 Journal Chem. Soc. Faraday
, 1997
"... In rare cases a neutral fullerene has an antibonding HOMO in simple Hu � ckel theory, thus constituting a counterexample to the mathematical conjecture that in all fullerenes the number of positive eigenvalues of the adjacency matrix is greater than or equal to the number of negatives. The examples ..."
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In rare cases a neutral fullerene has an antibonding HOMO in simple Hu � ckel theory, thus constituting a counterexample to the mathematical conjecture that in all fullerenes the number of positive eigenvalues of the adjacency matrix is greater than or equal to the number of negatives. The examples found here all have at least 628 carbon atoms and leave undisturbed the ruleofthumb prediction of electron deÐciency for the vast majority of fullerenes. In the classical deÐnition, a fullerene is an allcarbon molecule C in the form of a closed, trivalent, polyhedron with 12 penn tagonal faces and all other faces hexagonal. Its graph has therefore n vertices (atoms), 3n/2 edges (p bonds), (n/2 [ 10) hexagonal and 12 pentagonal faces (rings). Extension to other face recipes and topologies to give nonclassical fullerenes based on other trivalent polyhedra is possible.1h5 This paper deals with the numbers of bonding and antibonding n energy levels to be expected of a classical fullerene, and show that, contrary to mathematical conjecture and extensive numerical experience, a fullerene isomer may have an antibonding
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.
Independent sets and the prediction of additional patterns for higher fullerenes
 J. Chem. Soc. Perkin
, 1999
"... Under the assumptions that no two sp 3 carbon atoms are adjacent in the end product of bromination of a fullerene and that the residual π system is a closed shell, graph theory predicts maximum stoichiometries C 60Br 24, C 70Br 26, C 76Br 28, C 84Br 32 and rules out all but 58 of the ~10 23 addition ..."
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Under the assumptions that no two sp 3 carbon atoms are adjacent in the end product of bromination of a fullerene and that the residual π system is a closed shell, graph theory predicts maximum stoichiometries C 60Br 24, C 70Br 26, C 76Br 28, C 84Br 32 and rules out all but 58 of the ~10 23 addition patterns conceivable for these molecules.
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.
GraphDrawing Algorithms Geometries Versus Molecular Mechanics in Fullerenes
, 1996
"... The algorithms of KamadaKawai (KK) and FruchtermanReingold (FR) have been recently generalized (Pisanski et al., Croat. Chem. Acta 68 (1995) 283.) in order to draw molecular graphs in threedimensional space. The quality of KK and FR geometries is studied here by comparing them with the molecular ..."
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The algorithms of KamadaKawai (KK) and FruchtermanReingold (FR) have been recently generalized (Pisanski et al., Croat. Chem. Acta 68 (1995) 283.) in order to draw molecular graphs in threedimensional space. The quality of KK and FR geometries is studied here by comparing them with the molecular mechanics (MM) and the adjacency matrix eigenvectors (AME) algorithm geometries.
GoldbergCoxeter Construction for 3 and 4valent Plane Graphs
, 2004
"... We consider the GoldbergCoxeter construction GC k,l (G 0 ) (a generalization of a simplicial subdivision of the dodecahedron considered in [Gold37] and [Cox71]), which produces a plane graph from any 3 or 4valent plane graph for integer parameters k, l.Azigzag in a plane graph is a circuit of ed ..."
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We consider the GoldbergCoxeter construction GC k,l (G 0 ) (a generalization of a simplicial subdivision of the dodecahedron considered in [Gold37] and [Cox71]), which produces a plane graph from any 3 or 4valent plane graph for integer parameters k, l.Azigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face; a central circuit in a 4valent plane graph G is a circuit of edges, such that no two consecutive edges belong to the same face. We study the zigzag (or central circuit) structure of the resulting graph using the algebraic formalism of the moving group,the(k, l)product and a finite index subgroup of SL 2 (Z), whose elements preserve the above structure. We also study the intersection pattern of zigzags (or central circuits) of GC k,l (G 0 ) and consider its projections, obtained by removing all but one zigzags (or central circuits).
Lists of FaceRegular Polyhedra
, 1999
"... We introduce a new notion that connects the combinatorial concept of regularity with the geometrical notion of face transitivity. This new notion implies finiteness results in the case of bounded maximal face size. We give lists of structures for some classes and investigate polyhedra with constant ..."
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We introduce a new notion that connects the combinatorial concept of regularity with the geometrical notion of face transitivity. This new notion implies finiteness results in the case of bounded maximal face size. We give lists of structures for some classes and investigate polyhedra with constant vertex degrees and faces of only two sizes.
Construction of planar triangulations with minimum degree 5
, 1969
"... In this article we describe a method of constructing all simple triangulations of the sphere with minimum degree 5; equivalently, 3connected planar cubic graphs with girth 5. We also present the results of a computer program based on this algorithm, including counts of convex polytopes of minimum d ..."
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In this article we describe a method of constructing all simple triangulations of the sphere with minimum degree 5; equivalently, 3connected planar cubic graphs with girth 5. We also present the results of a computer program based on this algorithm, including counts of convex polytopes of minimum degree 5. Key words: planar triangulation, cubic graph, generation, fullerene
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...
Zigzags, railroads and knots in fullerenes
 J. CHEM. INF. COMPUT. SCI
, 2004
"... Two connections between fullerene structures and alternating knots are established. Knots may appear in two ways: from zigzags, i.e., circuits (possibly selfintersecting) of edges running alternately left and right at successive vertices, and from railroads, i.e., circuits (possibly selfintersecti ..."
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Two connections between fullerene structures and alternating knots are established. Knots may appear in two ways: from zigzags, i.e., circuits (possibly selfintersecting) of edges running alternately left and right at successive vertices, and from railroads, i.e., circuits (possibly selfintersecting) of edgesharing hexagonal faces, such that the shared edges occur in opposite pairs. A zknot fullerene has only a single zigzag, doubly covering all edges: in the range investigated (n e 74) examples are found for C34 and all Cn with n g 38, all chiral, belonging to groups C1, C2, C3, D3, orD5. Anrknot fullerene has a railroad corresponding to the projection of a nontrivial knot: examples are found for C52 (trefoil), C54 (figureofeight or Flemish knot), and, with isolated pentagons, at C96, C104, C108, C112, C114. Statistics on the occurrence of zknots and of zvectors of various kinds, zuniform, ztransitive, and zbalanced, are presented for trivalent polyhedra, general fullerenes, and isolatedpentagon fullerenes, along with examples with selfintersecting railroads and rknots. In a subset of zknot fullerenes, socalled minimal knots, the unique zigzag defines a specific Kekulé structure in which double bonds lie on lines of longitude and single bonds on lines of latitude of the approximate sphere defined by the polyhedron vertices.