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LCF Notation
In the mathematical field of graph theory, LCF notation or LCF code is a notation devised by Joshua Lederberg, and extended by H. S. M. Coxeter and Robert Frucht, for the representation of cubic graphs that contain a Hamiltonian cycle. The cycle itself includes two out of the three adjacencies for each vertex, and the LCF notation specifies how far along the cycle each vertex's third neighbor is. A single graph may have multiple different representations in LCF notation. Description In a Hamiltonian graph, the vertices can be arranged in a cycle, which accounts for two edges per vertex. The third edge from each vertex can then be described by how many positions clockwise (positive) or counter-clockwise (negative) it leads. The basic form of the LCF notation is just the sequence of these numbers of positions, starting from an arbitrarily chosen vertex and written in square brackets. The numbers between the brackets are interpreted modulo ''N'', where ''N'' is the number of vertic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nauru Graph LCF
Nauru ( or ; na, Naoero), officially the Republic of Nauru ( na, Repubrikin Naoero) and formerly known as Pleasant Island, is an island country and microstate in Oceania, in the Central Pacific. Its nearest neighbour is Banaba Island in Kiribati, about to the east. It further lies northwest of Tuvalu, northeast of Solomon Islands, east-northeast of Papua New Guinea, southeast of the Federated States of Micronesia and south of the Marshall Islands. With only a area, Nauru is the third-smallest country in the world behind Vatican City and Monaco, making it the smallest republic as well as the smallest island nation. Its population of about 10,000 is the world's second-smallest (not including colonies or overseas territories), after Vatican City. Settled by people from Micronesia circa 1000 BCE, Nauru was annexed and claimed as a colony by the German Empire in the late 19th century. After World War I, Nauru became a League of Nations mandate administered by Austra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecahedron
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120. Some dodecahedra have the same combinatorial structure as the regular dodecahedron (in terms of the graph formed by its vertices and edges), but their pentagonal faces are not regular: The pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the tetartoid has tetrahedral symmetry. The rhombic dodecahedron can be seen as a limiting case of the pyritohedron, and it has octahedral symmetry. The elongated dodecahedron and trapezo-rhombic dodecahedron variations, along with the rhombic dodecahedra, are space-filling. There ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Desargues Graph
In the mathematical field of graph theory, the Desargues graph is a distance-transitive, cubic graph with 20 vertices and 30 edges. It is named after Girard Desargues, arises from several different combinatorial constructions, has a high level of symmetry, is the only known non-planar cubic partial cube, and has been applied in chemical databases. The name "Desargues graph" has also been used to refer to a ten-vertex graph, the complement of the Petersen graph, which can also be formed as the bipartite half of the 20-vertex Desargues graph. Constructions There are several different ways of constructing the Desargues graph: *It is the generalized Petersen graph . To form the Desargues graph in this way, connect ten of the vertices into a regular decagon, and connect the other ten vertices into a ten-pointed star that connects pairs of vertices at distance three in a second decagon. The Desargues graph consists of the 20 edges of these two polygons together with an additional 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero-symmetric Graph
In the mathematics, mathematical field of graph theory, a zero-symmetric graph is a connected graph in which each vertex has exactly three incident edges and, for each two vertices, there is a unique graph automorphism, symmetry taking one vertex to the other. Such a graph is a vertex-transitive graph but cannot be an edge-transitive graph: the number of symmetries equals the number of vertices, too few to take every edge to every other edge. The name for this class of graphs was coined by R. M. Foster in a 1966 letter to Harold Scott MacDonald Coxeter, H. S. M. Coxeter. In the context of group theory, zero-symmetric graphs are also called graphical regular representations of their symmetry groups.. Examples The smallest zero-symmetric graph is a nonplanar graph with 18 vertices. Its LCF notation is [5,−5]9. Among planar graphs, the truncated cuboctahedral graph, truncated cuboctahedral and truncated icosidodecahedral graphs are also zero-symmetric. These examples are al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pappus Graph
In the mathematical field of graph theory, the Pappus graph is a bipartite 3- regular undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration. It is named after Pappus of Alexandria, an ancient Greek mathematician who is believed to have discovered the "hexagon theorem" describing the Pappus configuration. All the cubic distance-regular graphs are known; the Pappus graph is one of the 13 such graphs. The Pappus graph has rectilinear crossing number 5, and is the smallest cubic graph with that crossing number . It has girth 6, diameter 4, radius 4, chromatic number 2, chromatic index 3 and is both 3- vertex-connected and 3- edge-connected. It has book thickness 3 and queue number 2. The Pappus graph has a chromatic polynomial equal to: (x-1)x(x^-26x^+325x^-2600x^+14950x^-65762x^+229852x^-653966x^9+1537363x^8-3008720x^7+4904386x^6-6609926x^5+7238770x^4-6236975x^3+3989074x^2-1690406x+356509). The name "Pappus graph" has also been used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius–Kantor Graph
In the mathematical field of graph theory, the Möbius–Kantor graph is a symmetric bipartite cubic graph with 16 vertices and 24 edges named after August Ferdinand Möbius and Seligmann Kantor. It can be defined as the generalized Petersen graph ''G''(8,3): that is, it is formed by the vertices of an octagon, connected to the vertices of an eight-point star in which each point of the star is connected to the points three steps away from it. Möbius–Kantor configuration asked whether there exists a pair of polygons with ''p'' sides each, having the property that the vertices of one polygon lie on the lines through the edges of the other polygon, and vice versa. If so, the vertices and edges of these polygons would form a projective configuration. For ''p'' = 4 there is no solution in the Euclidean plane, but found pairs of polygons of this type, for a generalization of the problem in which the points and edges belong to the complex projective plane. That is, in Kantor's so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heawood Graph
Heawood is a surname. Notable people with the surname include: *Jonathan Heawood, British journalist *Percy John Heawood (1861–1955), British mathematician **Heawood conjecture **Heawood graph **Heawood number In mathematics, the Heawood number of a surface is an upper bound for the number of colors that suffice to color any graph embedded in the surface. In 1890 Heawood proved for all surfaces ''except'' the sphere that no more than : H(S)=\left\lfl ... See also * Heywood (surname) {{surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncated Tetrahedron
In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length. A deeper truncation, removing a tetrahedron of half the original edge length from each vertex, is called rectification. The rectification of a tetrahedron produces an octahedron. A ''truncated tetrahedron'' is the Goldberg polyhedron containing triangular and hexagonal faces. A ''truncated tetrahedron'' can be called a cantic cube, with Coxeter diagram, , having half of the vertices of the cantellated cube (rhombicuboctahedron), . There are two dual positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra. Area and volume The area ''A'' and the volume ''V'' of a truncated tetrahedron of edge length ''a'' are: :\begin A &= 7\sqrta^2 &&\appro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frucht Graph
In the mathematical field of graph theory, the Frucht graph is a cubic graph with 12 vertices, 18 edges, and no nontrivial symmetries. It was first described by Robert Frucht in 1939. The Frucht graph is a pancyclic, Halin graph with chromatic number 3, chromatic index 3, radius 3, and diameter 4. Like every Halin graph, the Frucht graph is polyhedral (planar and 3-vertex-connected) and Hamiltonian, with girth 3. Its independence number is 5. The Frucht graph can be constructed from the LCF notation: . Algebraic properties The Frucht graph is one of the five smallest cubic graphs possessing only a single graph automorphism, the identity (that is, every vertex can be distinguished topologically from every other vertex). Such graphs are called asymmetric (or identity) graphs. Frucht's theorem states that any group can be realized as the group of symmetries of a graph,. and a strengthening of this theorem also due to Frucht states that any group can be realized as the symmetr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Franklin Graph
In the mathematical field of graph theory, the Franklin graph is a 3-regular graph with 12 vertices and 18 edges. The Franklin graph is named after Philip Franklin, who disproved the Heawood conjecture on the number of colors needed when a two-dimensional surface is partitioned into cells by a graph embedding.Franklin, P. "A Six Color Problem." J. Math. Phys. 13, 363-379, 1934. The Heawood conjecture implied that the maximum chromatic number of a map on the Klein bottle should be seven, but Franklin proved that in this case six colors always suffice. (The Klein bottle is the only surface for which the Heawood conjecture fails.) The Franklin graph can be embedded in the Klein bottle so that it forms a map requiring six colors, showing that six colors are sometimes necessary in this case. This embedding is the Petrie dual of its embedding in the projective plane shown below. It is Hamiltonian and has chromatic number 2, chromatic index 3, radius 3, diameter 3 and girth 4. It is al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bidiakis Cube
In the mathematical field of graph theory, the bidiakis cube is a 3-regular graph with 12 vertices and 18 edges. Construction The bidiakis cube is a cubic Hamiltonian graph and can be defined by the LCF notation 6,4,-4sup>4. The bidiakis cube can also be constructed from a cube by adding edges across the top and bottom faces which connect the centres of opposite sides of the faces. The two additional edges need to be perpendicular to each other. With this construction, the bidiakis cube is a polyhedral graph, and can be realized as a convex polyhedron. Therefore, by Steinitz's theorem, it is a 3-vertex-connected simple planar graph.Branko Grünbaum, '' Convex Polytopes'', 2nd edition, prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler, 2003, , , 466pp. Algebraic properties The bidiakis cube is not a vertex-transitive graph and its full automorphism group is isomorphic to the dihedral group of order 8, the group of symmetries of a square, including both rotations a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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