Antiprism

Abstract: In this paper, we calculate the number of spanning trees in prism and antiprism graphs corresponding to the skeleton of a prism and an antiprism. By the electrically equivalent transformations and rules of weighted generating function, we obtain a relationship for the weighted number of spanning trees at the successive two generations. Using the knowledge of difference equations, we derive the analytical expressions for enumeration of spanning trees. In addition, we again calculate the number of spanning trees in Apollonian networks, which shows that this method is simple and effective. Finally we compare the entropy of our networks with other studied networks and find that the entropy of the antiprism graph is larger.

Abstract: Divided Spheres Working with Spheres Making a Point An Arbitrary Number Symmetry and Polyhedral Designs Spherical Workbenches Detailed Designs Other Ways to Use Polyhedra Summary Additional Resources Bucky's Dome Synergetic Geometry Dymaxion Projection Cahill and Waterman Projections Vector Equilibrium Icosa's The First Dome NC State and Skybreak Carolina Ford Rotunda Dome Marines in Raleigh University Circuit Radomes Kaiser's Domes Union Tank Car Covering Every Angle Summary Additional Resources Putting Spheres to Work Tammes Problem Spherical Viruses Celestial Catalogs Sudbury Neutrino Observatory Climate Models and Weather Prediction Cartography Honeycombs for Supercomputers Fish Farming Virtual Reality Modeling Spheres Dividing Golf Balls Spherical Throwable Panoramic Camera Hoberman's MiniSphere Rafiki's Code World Art and Expression Additional Resources Circular Reasoning Lesser and Great Circles Geodesic Subdivision Circle Poles Arc and Chord Factors Where Are We? Altitude-Azimuth Coordinates Latitude and Longitude Coordinates Spherical Trips Loxodromes Separation Angle Latitude Sailing Longitude Spherical Coordinates Cartesian Coordinates rho, phi, lambda Coordinates Spherical Polygons Excess and Defect Summary Additional Resources Distributing Points Covering Packing Volume Summary Additional Resources Polyhedral Frameworks What Is a Polyhedron? Platonic Solids Symmetry Archimedean Solids Additional Resources Golf Ball Dimples Icosahedral Balls Octahedral Balls Tetrahedral Balls Bilateral Symmetry Subdivided Areas Dimple Graphics Summary Additional Resources Subdivision Schemas Geodesic Notation Triangulation Number Frequency and Harmonics Grid Symmetry Class I: Alternates and Ford Class II: Triacon Class III: Skew Covering the Whole Sphere Additional Resources Comparing Results Kissing-Touching Sameness or Nearly So Triangle Area Face Acuteness Euler Lines Parts and T . 257 Convex Hull Spherical Caps Stereograms Face Orientation King Icosa Summary Additional Resources Computer-Aided Design A Short History CATIA Octet Truss Connector Spherical Design Three Class II Triacon Designs Panel Sphere Class II Strut Sphere Class II Parabolic Stellations Class I Ford Shell 31 Great Circles Class III Skew Additional Resources Advanced CAD Techniques Reference Models An Architectural Example Spherical Reference Models Prepackaged Reference and Assembly Models Local Axis Systems Assembly Review Design-in-Context Associative Geometry Design-in-Context versus Constraints Mirrored Enantiomorphs Power Copy Power Copy Prototype Macros Publication Data Structures CAD Alternatives: Stella and Antiprism Antiprism Summary Additional Resources Spherical Trigonometry Basic Trigonometric Functions The Core Theorems Law of Cosines Law of Sines Right Triangles Napier's Rule Using Napier's Rule on Oblique Triangles Polar Triangles Additional Resources Stereographic Projection Points on a Sphere Stereographic Properties A History of Diverse Uses The Astrolabe Crystallography and Geology Cartography Projection Methods Great Circles Lesser Circles Wulff Net Polyhedra Stereographics Polyhedra as Crystals Metrics and Interpretation Projecting Polyhedra Octahedron Tetrahedron Geodesic Stereographics Spherical Icosahedron Summary Additional Resources Geodesic Math Class I: Alternates and Fords Class II: Triacon Class III: Skew Characteristics of Triangles Storing Grid Points Additional Resources Schema Coordinates Coordinates for Class I: Alternates and Ford Coordinates for Class II: Triacon Coordinates for Class III: Skew Coordinate Rotations Rotation Concepts Direction and Sequences Simple Rotations Reflections Antipodal Points Compound Rotations Rotation around an Arbitrary Axis Polyhedra and Class Rotation Sequences Icosahedron Classes I and III Icosahedron Class Octahedron Classes I and III Octahedron Class Tetrahedron Classes I and III Tetrahedron Class Dodecahedron Class Cube Class Implementing Rotations Using Matrices Rotation Algorithms An Example Summary Additional Resources

Abstract: This paper unifies the following ideas for the study of chirality polynomials in transitive skeletons: (1) Generalization of chirality to permutation groups not corresponding to three-dimensional symmetry point groups leading to the concepts of signed permutation groups and their signed subgroups; (2) Determination of the total dimension of the chiral ligand partitions through the Frobenius reciprocity theorem; (3) Determination of signed permutation groups, not necessarily corresponding to three-dimensional point groups, of which a given ligand partition is a maximum symmetry chiral ligand partition by the Ruch-Schonhofer partial ordering, thereby allowing the determination of corresponding chirality polynomials depending only upon differences between ligand parameters; such permutation groups having the point group as a signed subgroup relate to qualitative completeness. In the case of transitive permutation groups on four sites, the tetrahedron and polarized square each have only one chiral ligand partition, but the allene and polarized rectangle skeletons each have two chiral ligand partitions related to their being signed subgroups of the tetrahedron and polarized square, respectively. The single transitive permutation group on five sites, the polarized pentagon, has a degenerate chiral ligand partition related to its being a signed subgroup of a metacyclic group with 20 elements. The octahedron has two chiral ligand partitions, both of degree six; a qualitatively complete chirality polynomial is therefore homogeneous of degree six. The cyclopropane (or trigonal prism or trigonal antiprism) skeleton is a signed subgroup of both the octahedron and a twist group of order 36; two of its six chiral ligand partitions come from the octahedron and two more from the twist group. The polarized hexagon is a signed subgroup of the same twist group but not of the octahedron and thus has a different set of six chiral ligand partitions than the cyclopropane skeleton. Two of its six chiral ligand partitions come from the above twist group of order 36 and two more from a signed permutation group of order 48 derived from the P3[P 2] wreath product group with a different assignment of positive and negative operations than the octahedron.