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75
GraphTheoretic Independence as a Predictor of Fullerene Stability
, 2003
"... The independence number of the graph of a fullerene, the size of the largest set of vertices such that no two are adjacent (corresponding to the largest set of atoms of the molecule, no pair of which are bonded), appears to be a useful selector in identifying stable fullerene isomers. The experiment ..."
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Cited by 9 (2 self)
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The independence number of the graph of a fullerene, the size of the largest set of vertices such that no two are adjacent (corresponding to the largest set of atoms of the molecule, no pair of which are bonded), appears to be a useful selector in identifying stable fullerene isomers. The experimentally characterized isomers with 60, 70 and 76 atoms uniquely minimize this number among the classes of possible structures with, respectively, 60, 70 and 76 atoms. Other experimentally characterized isomers also rank extremely low with respect to this invariant. These findings were initiated by a conjecture of the computer program Graffiti.
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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Cited by 6 (1 self)
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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.
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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Cited by 5 (2 self)
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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.
Fullerene Graphs With More Negative Than Positive Eigenvalues; The Exceptions That Prove The Rule of Electron De
 Journal Chem. Soc. Faraday
, 1997
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Cyclic edgecuts in fullerene graphs
 J. Math. Chem
"... In this paper, we study cyclic edgecuts in fullerene graphs. First, we show that the cyclic edgecuts of a fullerene graph can be constructed from its trivial cyclic 5 and 6edgecuts using three basic operations. This result immediatelly implies the fact that fullerene graphs are cyclically 5edg ..."
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Cited by 4 (1 self)
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In this paper, we study cyclic edgecuts in fullerene graphs. First, we show that the cyclic edgecuts of a fullerene graph can be constructed from its trivial cyclic 5 and 6edgecuts using three basic operations. This result immediatelly implies the fact that fullerene graphs are cyclically 5edgeconnected. Next, we characterize a class of nanotubes as the only fullerene graphs with nontrivial cyclic 5edgecuts. A similar result is also given for cyclic 6edgecuts of fullerene graphs. 1
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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Cited by 4 (3 self)
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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.
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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Cited by 4 (2 self)
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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.
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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Cited by 4 (3 self)
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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).
Faceregular bifaced polyhedra
, 2001
"... Call bifaced any kvalent polyhedron, whose faces are pa agons and pb bgons only, where 36a¡b,0¡pa, 06pb. We consider the case b62k=(k − 2) covering applications; so either k =36a¡b66, or (k; a; b; pa) = (4; 3; 4; 8). Call such a polyhedron aRi (resp., bRj) if each of its agonal (bgonal) faces ..."
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Cited by 3 (2 self)
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Call bifaced any kvalent polyhedron, whose faces are pa agons and pb bgons only, where 36a¡b,0¡pa, 06pb. We consider the case b62k=(k − 2) covering applications; so either k =36a¡b66, or (k; a; b; pa) = (4; 3; 4; 8). Call such a polyhedron aRi (resp., bRj) if each of its agonal (bgonal) faces is adjacent to exactly iagonal (resp., jbgonal) faces. The preferable (i.e., with isolated pentagons) fullerenes are the case aR0 for (k; a; b) = (3; 5; 6). We classify all
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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Cited by 3 (1 self)
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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