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Nauru Graph
In the mathematics, mathematical field of graph theory, the Nauru graph is a symmetric graph, symmetric bipartite graph, bipartite cubic graph with 24 vertices and 36 edges. It was named by David Eppstein after the twelve-pointed star in the flag of Nauru. It has chromatic number 2, chromatic index 3, diameter 4, radius 4 and girth 6.Marston Conder, Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002. It is also a 3-k-vertex-connected graph, vertex-connected and 3-k-edge-connected graph, edge-connected graph. It has book thickness 3 and queue number 2. The Nauru graph requires at least eight crossings in any drawing of it in the plane. It is one of three non-isomorphic graphs tied for being the smallest cubic graph that requires eight crossings. Another of these three graphs is the McGee graph, also known as the (3-7)-Cage (graph theory), cage. Construction The Nauru graph is hamiltonian graph, Hamiltonian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nauru Graph
In the mathematics, mathematical field of graph theory, the Nauru graph is a symmetric graph, symmetric bipartite graph, bipartite cubic graph with 24 vertices and 36 edges. It was named by David Eppstein after the twelve-pointed star in the flag of Nauru. It has chromatic number 2, chromatic index 3, diameter 4, radius 4 and girth 6.Marston Conder, Conder, M. and Dobcsányi, P. "Trivalent Symmetric Graphs Up to 768 Vertices." J. Combin. Math. Combin. Comput. 40, 41-63, 2002. It is also a 3-k-vertex-connected graph, vertex-connected and 3-k-edge-connected graph, edge-connected graph. It has book thickness 3 and queue number 2. The Nauru graph requires at least eight crossings in any drawing of it in the plane. It is one of three non-isomorphic graphs tied for being the smallest cubic graph that requires eight crossings. Another of these three graphs is the McGee graph, also known as the (3-7)-Cage (graph theory), cage. Construction The Nauru graph is hamiltonian graph, Hamiltonian ... [...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 same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge Philosophical Society
The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of law, theology and medicine. The society was granted a royal charter by King William IV in 1832. The society is governed by an elected council of senior academics, which is chaired by the Society's President, according to a set of statutes. The society has published several scientific journals, including ''Biological Reviews'' (established 1926) and ''Mathematical Proceedings of the Cambridge Philosophical Society'' (formerly entitled ''Proceedings of the Cambridge Philosophical Society'', published since 1843). ''Transactions of the Cambridge Philosophical Society'' was published between 1821–1928, but was then discontinued. History The society was founded in 1819 by Edward Clarke, Adam Sedgwick and John Stevens Henslow, and is Cambri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance-transitive Graph
In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices and at any distance , and any other two vertices and at the same distance, there is an automorphism of the graph that carries to and to . Distance-transitive graphs were first defined in 1971 by Norman L. Biggs and D. H. Smith. A distance-transitive graph is interesting partly because it has a large automorphism group. Some interesting finite groups are the automorphism groups of distance-transitive graphs, especially of those whose diameter is 2. Examples Some first examples of families of distance-transitive graphs include: * The Johnson graphs. * The Grassmann graphs. * The Hamming Graphs. * The folded cube graphs. * The square rook's graphs. * The hypercube graphs. * The Livingstone graph. Classification of cubic distance-transitive graphs After introducing them in 1971, Biggs and Smith showed that there are only 12 finite trivalent ... [...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 representatio ... [...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 s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dodecagon
In geometry, a dodecagon or 12-gon is any twelve-sided polygon. Regular dodecagon A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational symmetry of order 12. A regular dodecagon is represented by the Schläfli symbol and can be constructed as a truncated hexagon, t, or a twice-truncated triangle, tt. The internal angle at each vertex of a regular dodecagon is 150°. Area The area of a regular dodecagon of side length ''a'' is given by: :\begin A & = 3 \cot\left(\frac \right) a^2 = 3 \left(2+\sqrt \right) a^2 \\ & \simeq 11.19615242\,a^2 \end And in terms of the apothem ''r'' (see also inscribed figure), the area is: :\begin A & = 12 \tan\left(\frac\right) r^2 = 12 \left(2-\sqrt \right) r^2 \\ & \simeq 3.2153903\,r^2 \end In terms of the circumradius ''R'', the area is: :A = 6 \sin\left(\frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Petersen Graph
In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. The generalized Petersen graph family was introduced in 1950 by H. S. M. Coxeter and was given its name in 1969 by Mark Watkins. Definition and notation In Watkins' notation, ''G''(''n'', ''k'') is a graph with vertex set :\ and edge set :\ where subscripts are to be read modulo ''n'' and ''k'' < ''n''/2. Some authors use the notation ''GPG''(''n'', ''k''). Coxeter's notation for the same graph would be + , a combination of the Schläfli symbols for the regular ''n''-gon and star polygon from which the graph is formed. The Petersen graph itself is ''G''(5, 2) or + . Any generalized Petersen graph can also be constructed from a voltage graph with two vertices, two self-loops, and one other ed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Hamiltonian Graph
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. Determining whether such paths and cycles exist in graphs (the Hamiltonian path problem and Hamiltonian cycle problem) are NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as ''Hamilton's puzzle'', which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cage (graph Theory)
In the mathematical area of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an is defined to be a graph in which each vertex has exactly neighbors, and in which the shortest cycle has length exactly . An is an with the smallest possible number of vertices, among all . A is often called a . It is known that an exists for any combination of and . It follows that all exist. If a Moore graph exists with degree and girth , it must be a cage. Moreover, the bounds on the sizes of Moore graphs generalize to cages: any cage with odd girth must have at least :1+r\sum_^(r-1)^i vertices, and any cage with even girth must have at least :2\sum_^(r-1)^i vertices. Any with exactly this many vertices is by definition a Moore graph and therefore automatically a cage. There may exist multiple cages for a given combination of and . For instance there are three nonisomorphic , each with 70 vertices: the Balaban 10-cage, the Ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
McGee Graph
In the mathematical field of graph theory, the McGee graph or the (3-7)-cage is a 3-regular graph with 24 vertices and 36 edges. The McGee graph is the unique (3,7)-cage (the smallest cubic graph of girth 7). It is also the smallest cubic cage that is not a Moore graph. First discovered by Sachs but unpublished, the graph is named after McGee who published the result in 1960. Then, the McGee graph was proven the unique (3,7)-cage by Tutte in 1966. The McGee graph requires at least eight crossings in any drawing of it in the plane. It is one of three non-isomorphic graphs tied for being the smallest cubic graph that requires eight crossings. Another of these three graphs is the generalized Petersen graph , also known as the Nauru graph. The McGee graph has radius 4, diameter 4, chromatic number 3 and chromatic index 3. It is also a 3- vertex-connected and a 3- edge-connected graph. It has book thickness 3 and queue number 2. Algebraic properties The characteristic polynomial of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
