|
Foster Graph
In the mathematical field of graph theory, the Foster graph is a bipartite 3-regular graph with 90 vertices and 135 edges. The Foster graph is Hamiltonian and has chromatic number 2, chromatic index 3, radius 8, diameter 8 and girth 10. It is also a 3- vertex-connected and 3- edge-connected graph. It has queue number 2 and the upper bound on the book thickness is 4. All the cubic distance-regular graphs are known. The Foster graph is one of the 13 such graphs. It is the unique distance-transitive graph with intersection array . It can be constructed as the incidence graph of the partial linear space which is the unique triple cover with no 8-gons of the generalized quadrangle ''GQ''(2,2). It is named after R. M. Foster, whose ''Foster census'' of cubic symmetric graphs included this graph. The bipartite half of the Foster graph is a distance-regular graph and a locally linear graph. It is one of a finite number of such graphs with degree six. Algebraic properties The a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foster Graph
In the mathematical field of graph theory, the Foster graph is a bipartite 3-regular graph with 90 vertices and 135 edges. The Foster graph is Hamiltonian and has chromatic number 2, chromatic index 3, radius 8, diameter 8 and girth 10. It is also a 3- vertex-connected and 3- edge-connected graph. It has queue number 2 and the upper bound on the book thickness is 4. All the cubic distance-regular graphs are known. The Foster graph is one of the 13 such graphs. It is the unique distance-transitive graph with intersection array . It can be constructed as the incidence graph of the partial linear space which is the unique triple cover with no 8-gons of the generalized quadrangle ''GQ''(2,2). It is named after R. M. Foster, whose ''Foster census'' of cubic symmetric graphs included this graph. The bipartite half of the Foster graph is a distance-regular graph and a locally linear graph. It is one of a finite number of such graphs with degree six. Algebraic properties The a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Graph
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bipartite graph. Symmetry In 1932, Ronald M. Foster began collecting examples of cubic symmetric graphs, forming the start of the Foster census.. Many well-known individual graphs are cubic and symmetric, including the utility graph, the Petersen graph, the Heawood graph, the Möbius–Kantor graph, the Pappus graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs–Smith graph. W. T. Tutte classified the symmetric cubic graphs by the smallest integer number ''s'' such that each two oriented paths of length ''s'' can be mapped to each other by exactly one symmetry of the graph. He showed that ''s'' is at most 5, and provided examples of graphs with each possible ... [...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 corresponding eigenva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marston Conder
Marston Donald Edward Conder (born 9 September 1955) is a New Zealand mathematician, a Distinguished Professor of Mathematics at Auckland University,Staff directory listing entry Auckland U. Mathematics, retrieved 22 January 2013. and the former co-director of the New Zealand Institute of Mathematics and its Applications. His main research interests are in , , and their connections with each other. Education and career Conder was born in |
Locally Linear Graph
In graph theory, a locally linear graph is an undirected graph in which every edge belongs to exactly one triangle. Equivalently, for each vertex of the graph, its neighbors are each adjacent to exactly one other neighbor, so the neighbors can be paired up into an induced matching. Locally linear graphs have also been called locally matched graphs. Many constructions for locally linear graphs are known. Examples of locally linear graphs include the triangular cactus graphs, the line graphs of 3-regular triangle-free graphs, and the Cartesian products of smaller locally linear graphs. Certain Kneser graphs, and certain strongly regular graphs, are also locally linear. The question of how many edges locally linear graphs can have is one of the formulations of the Ruzsa–Szemerédi problem. Although dense graphs can have a number of edges proportional to the square of the number of vertices, locally linear graphs have a smaller number of edges, falling short of the square by at l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Half
In graph theory, the bipartite half or half-square of a bipartite graph is a graph whose vertex set is one of the two sides of the bipartition (without loss of generality, ) and in which there is an edge for each pair of vertices in that are at distance two from each other in . That is, in a more compact notation, the bipartite half is where the superscript 2 denotes the square of a graph and the square brackets denote an induced subgraph. Examples For instance, the bipartite half of the complete bipartite graph is the complete graph and the bipartite half of the hypercube graph is the halved cube graph. When is a distance-regular graph, its two bipartite halves are both distance-regular. For instance, the halved Foster graph is one of finitely many degree-6 distance-regular locally linear graphs. Representation and hardness Every graph is the bipartite half of another graph, formed by subdividing the edges of into two-edge paths. More generally, a representation of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foster Census
In the mathematical field of graph theory, a graph is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices and of , there is an automorphism :f : V(G) \rightarrow V(G) such that :f(u_1) = u_2 and f(v_1) = v_2. In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called -transitive or flag-transitive. By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex-transitive. Since the definition above maps one edge to another, a symmetric graph must also be edge-transitive. However, an edge-transitive graph need not be symmetric, since might map to , but not to . Star graphs are a simple example of being edge-transitive without being vertex-transitive or symmetric. As a further example, semi-symmetric graphs are edge-transitive and regular, but not vertex-transi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Quadrangle
In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles (yet containing many quadrangles). A generalized quadrangle is by definition a polar space of rank two. They are the with ''n'' = 4 and near 2n-gons with ''n'' = 2. They are also precisely the partial geometries pg(''s'',''t'',α) with α = 1. Definition A generalized quadrangle is an incidence structure (''P'',''B'',I), with I ⊆ ''P'' × ''B'' an incidence relation, satisfying certain axioms. Elements of ''P'' are by definition the ''points'' of the generalized quadrangle, elements of ''B'' the ''lines''. The axioms are the following: * There is an ''s'' (''s'' ≥ 1) such that on every line there are exactly ''s'' + 1 points. There is at most one point on two distinct lines. * There is a ''t'' (''t'' ≥ 1) such that through every point there are exactly ''t'' + 1 lines. There is at most one line through two distinct points. * For every point ''p'' not on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covering Space
A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete space D and for every x \in X an open neighborhood U \subset X, such that \pi^(U)= \displaystyle \bigsqcup_ V_d and \pi, _:V_d \rightarrow U is a homeomorphism for every d \in D . Often, the notion of a covering is used for the covering space E as well as for the map \pi : E \rightarrow X. The open sets V_ are called sheets, which are uniquely determined up to a homeomorphism if U is connected. For each x \in X the discrete subset \pi^(x) is called the fiber of x. The degree of a covering is the cardinality of the space D. If E is path-connected, then the covering \pi : E \rightarrow X is denoted as a path-connected covering. Examples * For every topological space X there exists the covering \pi:X \rightarrow X with \pi(x)=x, which is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Linear Space
A partial linear space (also semilinear or near-linear space) is a basic incidence structure in the field of incidence geometry, that carries slightly less structure than a linear space. The notion is equivalent to that of a linear hypergraph. Definition Let S=(,, \textbf) an incidence structure, for which the elements of are called ''points'' and the elements of are called ''lines''. ''S'' is a partial linear space, if the following axioms hold: * any line is incident with at least two points * any pair of distinct points is incident with at most one line If there is a unique line incident with every pair of distinct points, then we get a linear space. Properties The De Bruijn–Erdős theorem shows that in any finite linear space S=(,, \textbf) which is not a single point or a single line, we have , \mathcal, \leq , \mathcal, . Examples * Projective space * Affine space * Polar space * Generalized quadrangle * Generalized polygon * Near polygon References * . *Lynn Bat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incidence Graph
In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.. See in particulap. 181 From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line. They are named for Friedrich Wilhelm Levi, who wrote about them in 1942. The Levi graph of a system of points and lines usually has girth at least six: Any 4- cycles would correspond to two lines through the same two points. Conversely any bipartite graph with girth at least six can be viewed as the Levi graph of an abstract incidence structure. Levi graphs of configurations are biregular, and every biregular graph with girth at least six can be viewed as the Levi graph of an abstract configuration.. Levi graphs may also be defined for other types of incidence structure, such as the incidences between points and planes i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Generalized Quadrangle")
