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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 Harold Scott MacDonald Coxeter, H. S. M. Coxeter and Robert Frucht, for the representation of cubic graphs that contain a Hamiltonian path, Hamiltonian cycle. The cycle itself includes two out of the three adjacencies for each Vertex (graph theory), 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 circular layout, 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 numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nauru Graph LCF
Nauru, officially the Republic of Nauru, formerly known as Pleasant Island, is an island country and microstate in the South Pacific Ocean. It lies within the Micronesia subregion of Oceania, with its nearest neighbour being Banaba (part of Kiribati) about to the east. With an area of only , Nauru is the List of countries and dependencies by area, third-smallest country in the world, larger than only Vatican City and Monaco, making it the smallest republic and island nation, as well as the smallest member state of the Commonwealth of Nations by area. Demographics of Nauru, Its population of about 10,800 is the world's List of countries and dependencies by population, third-smallest (not including colonies or overseas territories). Nauru is a member of the United Nations, the Commonwealth of Nations, and the Organisation of African, Caribbean and Pacific States. Settled by Micronesians circa 1000 Common Era, BCE, Nauru was annexation, annexed and claimed as a colony by the G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tetrahedron is the simplest of all the ordinary convex polytope, convex polyhedra. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean geometry, Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid (geometry), 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 net (polyhedron), nets. For any tetrahedron there exists a sphere (called th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecahedron
In geometry, a dodecahedron (; ) 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 Kepler–Poinsot polyhedron, 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, pyritohedron, a common crystal form in pyrite, has pyritohedral symmetry, while the #Tetartoid, 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 polyhedra, space-filling. ... [...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 ... [...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) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius–Kantor Graph
In the mathematics, mathematical field of graph theory, the Möbius–Kantor graph is a symmetric graph, symmetric bipartite graph, 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 (an octagram). 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 bel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heawood Graph
In the mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood. Combinatorial properties The graph is cubic, and all cycles in the graph have six or more edges. Every smaller cubic graph has shorter cycles, so this graph is the 6-cage, the smallest cubic graph of girth 6. It is a distance-transitive graph (see the Foster census) and therefore distance regular. Additions and Corrections to the book Distance-Regular Graphs (Brouwer, Cohen, Neumaier; Springer; 1989) There are 24 perfect matchings in the Heawood graph; for each matching, the set of edges not in the matching forms a Hamiltonian cycle. For instance, the figure shows the vertices of the graph placed on a cycle, with the internal diagonals of the cycle forming a matching. By subdividing the cycle edges into two matchings, we can partition the Heawood graph into three perfect matchings (that is, 3-color its edges) in eight different wa ... [...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 truncation (geometry), truncating all 4 vertices of a regular tetrahedron. Construction The truncated tetrahedron can be constructed from a regular tetrahedron by cutting all of its vertices off, a process known as Truncation (geometry), truncation. The resulting polyhedron has 4 equilateral triangles and 4 regular hexagons, 18 edges, and 12 vertices. With edge length 1, the Cartesian coordinates of the 12 vertices are points \bigl( , \pm\tfrac, \pm\tfrac \bigr) that have an even number of minus signs. Properties Given the edge length a . The surface area of a truncated tetrahedron A is the sum of 4 regular hexagons and 4 equilateral triangles' area, and its volume V is: \begin A &= 7\sqrta^2 &&\approx 12.124a^2, \\ V &= \tfrac\sqrta^3 &&\approx 2.711a^3. \end The dihedral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frucht Graph
In the mathematics, mathematical field of graph theory, the Frucht graph is a cubic graph with 12 Vertex (graph theory), vertices, 18 edges, and no nontrivial graph automorphism, symmetries. It was first described by Robert Frucht in 1949. The Frucht graph is a pancyclic graph, pancyclic, Halin graph with chromatic number 3, chromatic index 3, radius 3, and diameter 4. Like every Halin graph, the Frucht graph is polyhedral graph, polyhedral (planar graph, planar and k-vertex-connected graph, 3-vertex-connected) and Hamiltonian graph, Hamiltonian, with Girth (graph theory), girth 3. Its Glossary_of_graph_theory#Independence, 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 graph, asymmetric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Franklin Graph
Franklin may refer to: People and characters * Franklin (given name), including list of people and characters with the name * Franklin (surname), including list of people and characters with the name * Franklin (class), a member of a historical English social class Places * Franklin (crater), a lunar impact crater * Franklin County (other), in a number of countries * Mount Franklin (other), including Franklin Mountain Australia * Franklin, Tasmania, a township * Division of Franklin, federal electoral division in Tasmania * Division of Franklin (state), state electoral division in Tasmania * Franklin, Australian Capital Territory, a suburb in the Canberra district of Gungahlin * Franklin River, river of Tasmania * Franklin Sound, waterway of Tasmania Canada * District of Franklin, a former district of the Northwest Territories * Franklin, Quebec, a municipality in the Montérégie region * Rural Municipality of Franklin, Manitoba * Franklin, Manitoba, ... [...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 rotatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
