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17
Construction of planar triangulations with minimum degree 5
"... 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.
Computers and Discovery in Algebraic Graph Theory
 Edinburgh, 2001), Linear Algebra Appl
, 2001
"... We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory. ..."
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Cited by 4 (0 self)
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We survey computers systems which help to obtain and sometimes provide automatically conjectures and refutations in algebraic graph theory.
The generation of fullerenes
 J. Chem. Inf. Model
"... We describe an efficient new algorithm for the generation of fullerenes. Our implementation of this algorithm is more than 3.5 times faster than the previously fastest generator for fullerenes – fullgen – and the first program since fullgen to be useful for more than 100 vertices. We also note a pro ..."
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We describe an efficient new algorithm for the generation of fullerenes. Our implementation of this algorithm is more than 3.5 times faster than the previously fastest generator for fullerenes – fullgen – and the first program since fullgen to be useful for more than 100 vertices. We also note a programming error in fullgen that caused problems for 136 or more vertices. We tabulate the numbers of fullerenes and IPR fullerenes up to 400 vertices. We also check up to 316 vertices a conjecture of Barnette that cubic planar graphs with maximum face size 6 are hamiltonian and verify that the smallest counterexample to the spiral conjecture has 380 vertices. Note: this is the unedited version of our paper which was submitted and subsequently accepted for publication in Journal of Chemical Information and Modeling. The final edited and published version can be accessed at
Generating Fullerenes at Random
, 1996
"... In the present paper a method for generating fullerenes at random is presented. It is based on the well known StoneWales (SW) transformation. The method could be further generalised so that other trivalent polyhedra with prescribed properties are generated. 1. INTRODUCTION Fullerenes and other pu ..."
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In the present paper a method for generating fullerenes at random is presented. It is based on the well known StoneWales (SW) transformation. The method could be further generalised so that other trivalent polyhedra with prescribed properties are generated. 1. INTRODUCTION Fullerenes and other pure carbon cages remain a subject of rigorous research. The mechanism of fullerenes growth is still not fully understood although much has been learned 1 . In the present paper a method for generating fullerenes at random is presented. It is based on the well known StoneWales (SW) transformation 2; 3 and it has been successfully implemented as a part of the VEGA: a system for manipulating discrete mathematical structures. 4; 5 The method could be further generalised so that it is able to narrow the population of carbon cages with special properties. From a mathematical standpoint a fullerene is planar trivalent graph whose faces are pentagons and hexagons. It turns out that the number...
Tree Orbits under Permutation Group Action: Algorithm, Enumeration and Application to Viral Assembly
, 2009
"... This paper uses combinatorics and group theory to answer questions about the assembly of icosahedral viral shells. Although the geometric structure of the capsid (shell) is fairly well understood in terms of its constituent subunits, the assembly process is not. For the purpose of this paper, the ca ..."
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This paper uses combinatorics and group theory to answer questions about the assembly of icosahedral viral shells. Although the geometric structure of the capsid (shell) is fairly well understood in terms of its constituent subunits, the assembly process is not. For the purpose of this paper, the capsid is modeled by a polyhedron whose facets represent the monomers. The assembly process is modeled by a rooted tree, the leaves representing the facets of the polyhedron, the root representing the assembled polyhedron, and the internal vertices representing intermediate stages of assembly (subsets of facets). Besides its virological motivation, the enumeration of orbits of trees under the action of a finite group is of independent mathematical interest. If G is a finite group acting on a finite set X, then there is a natural induced action of G on the set TX of trees whose leaves are bijectively labeled by the elements of X. If G acts simply on X, then X : = Xn  = n · G, where n is the number of Gorbits in X. The basic combinatorial results in this paper are (1) a formula for the number of orbits of each size in the action of G on TXn, for every n, and (2) a simple algorithm to find the stabilizer of a tree τ ∈ TX in G that runs in linear time and does not need memory in addition to its input tree.
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.
Finding Fullerene Patches in Polynomial Time
, 907
"... Abstract. We consider the following question, motivated by the enumeration of fullerenes. A fullerene patch is a 2connected plane graph G in which inner faces have length 5 or 6, nonboundary vertices have degree 3, and boundary vertices have degree 2 or 3. The degree sequence along the boundary is ..."
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Abstract. We consider the following question, motivated by the enumeration of fullerenes. A fullerene patch is a 2connected plane graph G in which inner faces have length 5 or 6, nonboundary vertices have degree 3, and boundary vertices have degree 2 or 3. The degree sequence along the boundary is called the boundary code of G. We show that the question whether a given sequence S is a boundary code of some fullerene patch can be answered in polynomial time when such patches have at most five 5faces. We conjecture that our algorithm gives the correct answer for any number of 5faces, and sketch how to extend the algorithm to the problem of counting the number of different patches with a given boundary code. 1
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"... Recursive generation of simple planar quadrangulations with vertices of degree 3 and 4 ..."
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Recursive generation of simple planar quadrangulations with vertices of degree 3 and 4