|
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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Franklin Graph Hamiltonian
Franklin may refer to: People * Franklin (given name) * Franklin (surname) * Franklin (class), a member of a historical English social class Places 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, an unincorporated community in the Rural Municipality of Rosedale, Manitoba * Franklin Glacier Complex, a volcano in southwestern British Columbia * Franklin Range, a mountain range on Vancouver Island, British Columbia * Franklin River (Vancouver Island), British Columbia * Franklin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petrie Dual
In topological graph theory, the Petrie dual of an embedded graph (on a 2-manifold with all faces disks) is another embedded graph that has the Petrie polygons of the first embedding as its faces. The Petrie dual is also called the Petrial, and the Petrie dual of an embedded graph G may be denoted G^\pi. It can be obtained from a signed rotation system or ribbon graph representation of the embedding by twisting every edge of the embedding. Properties Like the usual dual graph, repeating the Petrie dual operation twice returns to the original surface embedding. Unlike the usual dual graph (which is an embedding of a generally different graph in the same surface) the Petrie dual is an embedding of the same graph in a generally different surface. Surface duality and Petrie duality are two of the six Wilson operations, and together generate the group of these operations. Regular polyhedra Applying the Petrie dual to a regular polyhedron produces a regular map. The number of skew ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids. Uniform truncation In general any polyhedron (or polytope) can also be truncated with a degree of freedom as to how deep the cut is, as shown in Conway polyhedron notation truncation operation. A special kind of truncation, usually implied, is a uniform truncation, a truncation operator applied to a regular polyhedron (or regular polytope) which creates a resulting uniform polyhedron (uniform polytope) with equal edge lengths. There are no degrees of freedom, and it represents a fixed geometric, just like the regular polyhedra. In general all single ringed uniform polytopes have a uniform truncation. For example, the icosidodecahedron, represented as Schläfli symbols r or \begin 5 \\ 3 \end, and Coxeter-Dynkin diagram or has a uniform truncation, the truncated icosidodeca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any base (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Motivation In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the correspondin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \mathrm_n defined over a finite set of n symbols consists of the permutations that can be performed on the n symbols. Since there are n! (n factorial) such permutation operations, the order (number of elements) of the symmetric group \mathrm_n is n!. Although symmetric groups can be defined on infinite sets, this article focuses on the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element ''g'' such that every other element of the group may be obtained by repeatedly applying the group operation to ''g'' or its inverse. Each element can be written as an integer power of ''g'' in multiplicative notation, or as an integer multiple of ''g'' in additive notation. This element ''g'' is called a '' generator'' of the group. Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order ''n'' is isomorphic to the additive group of Z/''n''Z, the integers modulo ''n''. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Direct Product
A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identity * Religious group (other), a group whose members share the same religious identity * Social group, a group whose members share the same social identity * Tribal group, a group whose members share the same tribal identity * Organization, an entity that has a collective goal and is linked to an external environment * Peer group, an entity of three or more people with similar age, ability, experience, and interest Social science * In-group and out-group * Primary, secondary, and reference groups * Social group * Collectives Science and technology Mathematics * Group (mathematics), a set together with a binary operation satisfying certain algebraic conditions Chemistry * Functional group, a group of atoms which provide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Automorphism
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity. Formally, an automorphism of a graph is a permutation of the vertex set , such that the pair of vertices form an edge if and only if the pair also form an edge. That is, it is a graph isomorphism from to itself. Automorphisms may be defined in this way both for directed graphs and for undirected graphs. The composition of two automorphisms is another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphism group of the graph. In the opposite direction, by Frucht's theorem, all groups can be represented as the automorphism group of a connected graph – indeed, of a cubic graph. Computational complexity Constructing the automorphism group is at least as difficult (in terms of its computational complexity) as solving the graph i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Graph
In graph theory, a perfect graph is a graph in which the chromatic number of every induced subgraph equals the order of the largest clique of that subgraph (clique number). Equivalently stated in symbolic terms an arbitrary graph G=(V,E) is perfect if and only if for all S\subseteq V we have \chi(G =\omega(G . The perfect graphs include many important families of graphs and serve to unify results relating colorings and cliques in those families. For instance, in all perfect graphs, the graph coloring problem, maximum clique problem, and maximum independent set problem can all be solved in polynomial time. In addition, several important min-max theorems in combinatorics, such as Dilworth's theorem, can be expressed in terms of the perfection of certain associated graphs. A graph G is 1-perfect if and only if \chi(G)=\omega(G). Then, G is perfect if and only if every induced subgraph of G is 1-perfect. Properties * By the perfect graph theorem, a graph G is perfect if an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-edge-connected Graph
In graph theory, a connected graph is -edge-connected if it remains connected whenever fewer than edges are removed. The edge-connectivity of a graph is the largest for which the graph is -edge-connected. Edge connectivity and the enumeration of -edge-connected graphs was studied by Camille Jordan in 1869. Formal definition Let G = (V, E) be an arbitrary graph. If subgraph G' = (V, E \setminus X) is connected for all X \subseteq E where , X, < k, then ''G'' is ''k''-edge-connected. The edge connectivity of is the maximum value ''k'' such that ''G'' is ''k''-edge-connected. The smallest set ''X'' whose removal disconnects ''G'' is a minimum cut in ''G''. The edge connectivity version of Menger's theorem provides an alternative and equivalent characterization, in terms o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-vertex-connected Graph
In graph theory, a connected graph is said to be -vertex-connected (or -connected) if it has more than vertices and remains connected whenever fewer than vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest for which the graph is -vertex-connected. Definitions A graph (other than a complete graph) has connectivity ''k'' if ''k'' is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them. Complete graphs are not included in this version of the definition since they cannot be disconnected by deleting vertices. The complete graph with ''n'' vertices has connectivity ''n'' − 1, as implied by the first definition. An equivalent definition is that a graph with at least two vertices is ''k''-connected if, for every pair of its vertices, it is possible to find ''k'' vertex-independent paths connecting these vertices; see Menger's theorem . This definition produces the sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Girth (graph Theory)
In graph theory, the girth of an undirected graph is the length of a shortest cycle contained in the graph. If the graph does not contain any cycles (that is, it is a forest), its girth is defined to be infinity. For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3. A graph with girth four or more is triangle-free. Cages A cubic graph (all vertices have degree three) of girth that is as small as possible is known as a - cage (or as a -cage). The Petersen graph is the unique 5-cage (it is the smallest cubic graph of girth 5), the Heawood graph is the unique 6-cage, the McGee graph is the unique 7-cage and the Tutte eight cage is the unique 8-cage. There may exist multiple cages for a given girth. For instance there are three nonisomorphic 10-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries–Wong graph. Image:Petersen1 tiny.svg, The Petersen graph has a girth of 5 Image:Heawood_Gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
