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Polytope Skeletons And Paths
 Handbook of Discrete and Computational Geometry (Second Edition ), chapter 20
"... INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent i ..."
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INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t
Inner Diagonals Of Convex Polytopes
, 1998
"... . An inner diagonal of a polytope P is a segment that joins two vertices of P and that lies, except for its ends, in P 's relative interior. The paper's main results are as follows: (a) Among all dpolytopes P having a given number v of vertices, the maximum number of inner diagonals is \Gamma v ..."
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. An inner diagonal of a polytope P is a segment that joins two vertices of P and that lies, except for its ends, in P 's relative interior. The paper's main results are as follows: (a) Among all dpolytopes P having a given number v of vertices, the maximum number of inner diagonals is \Gamma v 2 \Delta \Gamma dv + \Gamma d+1 2 \Delta ; when d 4 it is attained if and only if P is a stacked polytope. (b) Among all dpolytopes having a given number f of facets, the maximum number of inner diagonals is attained by (and, at least when d = 3 and f 6, only by) certain simple polytopes. (c) When d = 3, the maximum in (b) is determined for all f ; when f 14 it is 2f 2 \Gamma 21f + 64 and the unique associated pvector is 5 12 6 f \Gamma12 . (d) Among all simple 3polytopes with f facets, the minimum number of inner diagonals is f 2 \Gamma 9f + 20; when f 9 the unique associated pvector is 3 2 4 f \Gamma4 (f \Gamma 1) 2 and the unique associated combinatorial type i...
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.
An Eberhardlike theorem for pentagons and heptagons
"... Eberhard proved that for every sequence (pk), 3 ≤ k ≤ r, k ̸ = 5, 7 of nonnegative integers satisfying Euler’s formula ∑ k≥3 (6 − k)pk = 12, ..."
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Eberhard proved that for every sequence (pk), 3 ≤ k ≤ r, k ̸ = 5, 7 of nonnegative integers satisfying Euler’s formula ∑ k≥3 (6 − k)pk = 12,
The symmetries of cubic polyhedral graphs with face size no larger than 6
 MATCH COMMUNICATIONS IN MATHEMATICAL
, 2009
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UNIVERSITY OF MALAYA KUALA LUMPUR
, 2006
"... I would like to express my sincere thanks to my former supervisor, Associate Professor Dr. Thomas Bier, for his unlimited assistance and encouragement throughout the course of my study and for helping me to complete my research work on time. I would also like to express my sincere thanks to my prese ..."
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I would like to express my sincere thanks to my former supervisor, Associate Professor Dr. Thomas Bier, for his unlimited assistance and encouragement throughout the course of my study and for helping me to complete my research work on time. I would also like to express my sincere thanks to my present supervisor, Associate Professor Dr. Angelina Chin Yan Mui for taking on the task of supervising me after my former supervisor went on leave. Her guidance and valuable suggestions helped me a lot to complete my thesis. Many thanks also to the Head and all staff of the Institute of Mathematical Sciences, University of Malaya for their assistance in numerous ways. Finally, I wish to express my special thanks to my loving mother and dearest wife for their encouragement to successfully complete my MSc studies. Panchadcharam Elango, 2002. The main objective of this research is to find the different types of elliptic
PROBABILISTIC INDUCTIVE CLASSES OF GRAPHS
, 2006
"... Abstract. Models of complex networks are generally defined as graph stochastic processes in which edges and vertices are added or deleted over time to simulate the evolution of networks. Here, we define a unifying framework — probabilistic inductive classes of graphs — for formalizing and studying e ..."
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Abstract. Models of complex networks are generally defined as graph stochastic processes in which edges and vertices are added or deleted over time to simulate the evolution of networks. Here, we define a unifying framework — probabilistic inductive classes of graphs — for formalizing and studying evolution of complex networks. Our definition of probabilistic inductive class of graphs (PICG) extends the standard notion of inductive class of graphs (ICG) by imposing a probability space. A PICG is given by: (1) class B of initial graphs, the basis of PICG, (2) class R of generating rules, each with distinguished left element to which the rule is applied to obtain the right element, (3) probability distribution specifying how the initial graph is chosen from class B, (4) probability distribution specifying how the rules from class R are applied, and, finally, (5) probability distribution specifying how the left elements for every rule in class R are chosen. We point out that many of the existing models of growing networks can be cast as PICGs. We present how the well known model of growing networks — the preferential attachment model — can be studied as PICG. As an illustration we present results regarding the size, order, and degree sequence for PICG models of connected and 2connected graphs. 1.