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Shrikhande Graph
In the mathematics, mathematical field of graph theory, the Shrikhande graph is a Gallery of named graphs, named graph discovered by S. S. Shrikhande in 1959.. It is a strongly regular graph with 16 vertex (graph theory), vertices and 48 edge (graph theory), edges, with each vertex having degree (graph theory), degree 6. Every pair of nodes has exactly two other neighbors in common, whether the pair of nodes is connected or not. Construction The Shrikhande graph can be constructed as a Cayley graph. The vertex set is \mathbb_4 \times \mathbb_4. Two vertices are adjacent if and only if the difference is in \. Properties In the Shrikhande graph, any two vertices ''I'' and ''J'' have two distinct neighbors in common (excluding the two vertices ''I'' and ''J'' themselves), which holds true whether or not ''I'' is adjacent to ''J''. In other words, it is strongly regular graph, strongly regular and its parameters are: , i.e., \lambda = \mu = 2. This equality implies that the gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Graph
In the mathematical discipline of graph theory, the line graph of an undirected graph is another graph that represents the adjacencies between edges of . is constructed in the following way: for each edge in , make a vertex in ; for every two edges in that have a vertex in common, make an edge between their corresponding vertices in . The name line graph comes from a paper by although both and used the construction before this. Other terms used for the line graph include the covering graph, the derivative, the edge-to-vertex dual, the conjugate, the representative graph, and the θ-obrazom, as well as the edge graph, the interchange graph, the adjoint graph, and the derived graph., p. 71. proved that with one exceptional case the structure of a connected graph can be recovered completely from its line graph. Many other properties of line graphs follow by translating the properties of the underlying graph from vertices into edges, and by Whitney's theorem the same t ... [...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 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance-regular Graph
In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices and , the number of vertices at distance from and at distance from depends only upon , , and the distance between and . Every distance-transitive graph is distance-regular. Indeed, distance-regular graphs were introduced as a combinatorial generalization of distance-transitive graphs, having the numerical regularity properties of the latter without necessarily having a large automorphism group. Intersection arrays It turns out that a graph G of diameter d is distance-regular if and only if there is an array of integers \ such that for all 1 \leq j \leq d , b_j gives the number of neighbours of u at distance j+1 from v and c_j gives the number of neighbours of u at distance j - 1 from v for any pair of vertices u and v at distance j on G . The array of integers characterizing a distance-regular graph is known as its intersection array. Cos ... [...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] |
Dyck Graph
In the mathematical field of graph theory, the Dyck graph is a 3-regular graph with 32 vertices and 48 edges, named after Walther von Dyck. It is Hamiltonian with 120 distinct Hamiltonian cycles. It has chromatic number 2, chromatic index 3, radius 5, diameter 5 and girth 6. It is also a 3- vertex-connected and a 3- edge-connected graph. It has book thickness 3 and queue number 2. The Dyck graph is a toroidal graph, and the dual of its symmetric toroidal embedding is the Shrikhande graph, a strongly regular graph both symmetric and hamiltonian. Algebraic properties The automorphism group of the Dyck graph is a group of order 192. It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Dyck graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. According to the ''Foster census'', the Dyck graph, referenced as F32A, is the only cubic symmetric graph on 32 vertices. The char ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Map (graph Theory)
In mathematics, a regular map is a symmetric tessellation of a closed surface. More precisely, a regular map is a decomposition of a two-dimensional manifold (such as a sphere, torus, or real projective plane) into topological disks such that every flag (an incident vertex-edge-face triple) can be transformed into any other flag by a symmetry of the decomposition. Regular maps are, in a sense, topological generalizations of Platonic solids. The theory of maps and their classification is related to the theory of Riemann surfaces, hyperbolic geometry, and Galois theory. Regular maps are classified according to either: the genus and orientability of the supporting surface, the underlying graph, or the automorphism group. Overview Regular maps are typically defined and studied in three ways: topologically, group-theoretically, and graph-theoretically. Topological approach Topologically, a map is a 2-cell decomposition of a compact connected 2-manifold. The genus g, of a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toroidal Graph
In the mathematical field of graph theory, a toroidal graph is a graph that can be embedded on a torus. In other words, the graph's vertices can be placed on a torus such that no edges cross. Examples Any graph that can be embedded in a plane can also be embedded in a torus. A toroidal graph of genus 1 can be embedded in a torus but not in a plane. The Heawood graph, the complete graph K7 (and hence K5 and K6), the Petersen graph (and hence the complete bipartite graph K3,3, since the Petersen graph contains a subdivision of it), one of the Blanuša snarks, and all Möbius ladders are toroidal. More generally, any graph with crossing number 1 is toroidal. Some graphs with greater crossing numbers are also toroidal: the Möbius–Kantor graph, for example, has crossing number 4 and is toroidal. Properties Any toroidal graph has chromatic number at most 7. The complete graph K7 provides an example of a toroidal graph with chromatic number 7. Any triangle-free toroidal graph h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andries Brouwer
Andries Evert Brouwer (born 1951) is a Dutch mathematician and computer programmer, Professor Emeritus at Eindhoven University of Technology (TU/e). He is known as the creator of the greatly expanded 1984 to 1985 versions of the roguelike computer game ''Hack'' that formed the basis for ''NetHack''. He is also a Linux kernel hacker. He is sometimes referred to by the handle ''aeb''. Biography Born in Amsterdam, Brouwer attended the gymnasium, and obtained his MSc in mathematics at the University of Amsterdam in 1971. In 1976 he received his Ph.D. in mathematics from Vrije Universiteit with a thesis entitled "Treelike Spaces and Related Topological Spaces", under the supervision of Maarten Maurice and Pieter Baayen, both of whom were in turn students of Johannes de Groot. In 2004 he received an honorary doctorate from Aalborg University. After graduation Brouwer started his academic career at the Mathematisch Centrum, later Centrum Wiskunde & Informatica. From 1986 to 2012 he w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a spindle torus. If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a ''toroid'', as in a square toroid. Real-world objects that approximate a torus of revolution include swim rings, inner tubes and ringette rings. Eyeglass lenses that combine spherical and cylindrical correction are toric lenses. A torus should not be confused with a '' solid torus'', which is formed by r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangulation (topology)
In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation has various uses in different branches of mathematics, for instance in algebraic topology, in complex analysis or in modeling. Motivation On the one hand, it is sometimes useful to forget about superfluous information of topological spaces: The replacement of the original spaces with simplicial complexes may help to recognize crucial properties and to gain a better understanding of the considered object. On the other hand, simplicial complexes are objects of combinatorial character and therefore one can assign them quantities rising from their combinatorial pattern, for instance, the Euler characteristic. Triangulation allows now to assign such quantities to topological spaces. Investigations concerning the exist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-skeleton
In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the simplices of (resp. cells of ) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the . These subspaces increase with . The is a discrete space, and the a topological graph. The skeletons of a space are used in obstruction theory, to construct spectral sequences by means of filtrations, and generally to make inductive arguments. They are particularly important when has infinite dimension, in the sense that the do not become constant as In geometry In geometry, a of P (functionally represented as skel''k''(''P'')) consists of all elements of dimension up to ''k''. For example: : skel0(cube) = 8 vertices : skel1(cube) = 8 vertices, 12 edges : skel2(cube) = 8 vertices, 12 edges, 6 square faces For simplicial sets The above def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
