|
M22 Graph
The M22 graph, also called the Mesner graph or Witt graph is the unique strongly regular graph with parameters (77, 16, 0, 4). Brouwer, Andries E. “M22 Graph.” Technische Universiteit Eindhoven, http://www.win.tue.nl/~aeb/graphs/M22.html. Accessed 29 May 2018. It is constructed from the Steiner system (3, 6, 22) by representing its 77 blocks as vertices and joining two vertices iff they have no terms in common or by deleting a vertex and its neighbors from the Higman–Sims graph.Weisstein, Eric W. “M22 Graph.” MathWorld, http://mathworld.wolfram.com/M22Graph.html. Accessed 29 May 2018.Vis, Timothy. “The Higman–Sims Graph.” University of Colorado Denver, http://math.ucdenver.edu/~wcherowi/courses/m6023/tim.pdf. Accessed 29 May 2018. For any term, the family of blocks that contain that term forms an independent set in this graph, with 21 vertices. In a result analogous to the Erdős–Ko–Rado theorem (which can be formulated in terms of independent sets in Knese ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Group M22
In the area of modern algebra known as group theory, the Mathieu group ''M22'' is a sporadic simple group of Order (group theory), order : 27325711 = 443520 : ≈ 4. History and properties ''M22'' is one of the 26 sporadic groups and was introduced by . It is a 3-fold transitive permutation group on 22 objects. The Schur multiplier of M22 is cyclic of order 12, and the outer automorphism group has order 2. There are several incorrect statements about the 2-part of the Schur multiplier in the mathematical literature. incorrectly claimed that the Schur multiplier of M22 has order 3, and in a correction incorrectly claimed that it has order 6. This caused an error in the title of the paper announcing the discovery of the Janko group J4. showed that the Schur multiplier is in fact cyclic of order 12. calculated the 2-part of all the cohomology of M22. Representations M22 has a 3-transitive permutation representation on 22 points, with point stabilizer th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Regular Graph
In graph theory, a strongly regular graph (SRG) is defined as follows. Let be a regular graph with vertices and degree . is said to be strongly regular if there are also integers and such that: * Every two adjacent vertices have common neighbours. * Every two non-adjacent vertices have common neighbours. The complement of an is also strongly regular. It is a . A strongly regular graph is a distance-regular graph with diameter 2 whenever μ is non-zero. It is a locally linear graph whenever . Etymology A strongly regular graph is denoted an srg(''v'', ''k'', λ, μ) in the literature. By convention, graphs which satisfy the definition trivially are excluded from detailed studies and lists of strongly regular graphs. These include the disjoint union of one or more equal-sized complete graphs, and their complements, the complete multipartite graphs with equal-sized independent sets. Andries Brouwer and Hendrik van Maldeghem (see #References) use an alternate but fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Technische Universiteit Eindhoven
The Eindhoven University of Technology ( nl, Technische Universiteit Eindhoven), abbr. TU/e, is a public technical university in the Netherlands, located in the city of Eindhoven. In 2020–21, around 14,000 students were enrolled in its BSc and MSc programs and around 1350 students were enrolled in its PhD and PDEng programs. In 2021, the TU/e employed around 3900 people. Eindhoven University of Technology has been ranked in the top 200 universities in three major ranking systems. The 2019 QS World University Rankings place Eindhoven 99th in the world, 34th in Europe, and 3rd in the Netherlands. TU/e is the Dutch member of thEuroTech Universities Alliance a strategic partnership of universities of science & technology in Europe: Technical University of Denmark (DTU), École Polytechnique Fédérale de Lausanne (EPFL), École Polytechnique (L’X), The Technion, Eindhoven University of Technology (TU/e), and Technical University of Munich (TUM). History The Eindhoven Uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steiner System
250px, thumbnail, The Fano plane is a Steiner triple system S(2,3,7). The blocks are the 7 lines, each containing 3 points. Every pair of points belongs to a unique line. In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and ''t'' = 2 or (recently) ''t'' ≥ 2. A Steiner system with parameters ''t'', ''k'', ''n'', written S(''t'',''k'',''n''), is an ''n''-element set ''S'' together with a set of ''k''-element subsets of ''S'' (called blocks) with the property that each ''t''-element subset of ''S'' is contained in exactly one block. In an alternate notation for block designs, an S(''t'',''k'',''n'') would be a ''t''-(''n'',''k'',1) design. This definition is relatively new. The classical definition of Steiner systems also required that ''k'' = ''t'' + 1. An S(2,3,''n'') was (and still is) called a ''Steiner triple'' (or ''triad'') ''system'', while an S(3,4,''n'') is called a ''Steiner quad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higman–Sims Graph
In mathematical graph theory, the Higman–Sims graph is a 22-regular graph, regular undirected graph with 100 vertices and 1100 edges. It is the unique strongly regular graph srg(100,22,0,6), where no neighboring pair of vertices share a common neighbor and each non-neighboring pair of vertices share six common neighbors. It was first constructed by and rediscovered in 1968 by Donald G. Higman and Charles C. Sims as a way to define the Higman–Sims group, a subgroup of Index of a subgroup, index two in the group of automorphisms of the Higman–Sims graph. Construction From M22 graph Take the M22 graph, a strongly regular graph srg(77,16,0,4) and augment it with 22 new vertices corresponding to the points of S(3,6,22), each block being connected to its points, and one additional vertex ''C'' connected to the 22 points. From Hoffman–Singleton graph There are 100 independent set (graph theory), independent sets of size 15 in the Hoffman–Singleton graph. Create a new graph w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent Set (graph Theory)
In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S of vertices such that for every two vertices in S, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in S. A set is independent if and only if it is a clique in the graph's complement. The size of an independent set is the number of vertices it contains. Independent sets have also been called "internally stable sets", of which "stable set" is a shortening. A maximal independent set is an independent set that is not a proper subset of any other independent set. A maximum independent set is an independent set of largest possible size for a given graph G. This size is called the independence number of ''G'' and is usually denoted by \alpha(G). The optimization problem of finding such a set is called the maximum independent set problem. It is a strongly NP-hard problem. As such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Ko–Rado Theorem
In mathematics, the Erdős–Ko–Rado theorem limits the number of sets in a family of sets for which every two sets have at least one element in common. Paul Erdős, Chao Ko, and Richard Rado proved the theorem in 1938, but did not publish it until 1961. It is part of the field of combinatorics, and one of the central results of The theorem applies to families of sets that all have the same and are all subsets of some larger set of size One way to construct a family of sets with these parameters, each two sharing an element, is to choose a single element to belong to all the subsets, and then form all of the subsets that contain the chosen element. The Erdős–Ko–Rado theorem states that when n is large enough for the problem to be nontrivial this construction produces the largest possible intersecting families. When n=2r there are other equally-large families, but for larger values of n only the families constructed in this way can be largest. The Erdős–Ko–Rado th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kneser Graph
In graph theory, the Kneser graph (alternatively ) is the graph whose vertices correspond to the -element subsets of a set of elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint. Kneser graphs are named after Martin Kneser, who first investigated them in 1956. Examples The Kneser graph is the complete graph on vertices. The Kneser graph is the complement of the line graph of the complete graph on vertices. The Kneser graph is the odd graph ; in particular is the Petersen graph (see top right figure). The Kneser graph , visualized on the right. Properties Basic properties The Kneser graph K(n,k) has \tbinom vertices. Each vertex has exactly \tbinom neighbors. The Kneser graph is vertex transitive and arc transitive. When k=2, the Kneser graph is a strongly regular graph, with parameters ( \tbinom, \tbinom, \tbinom, \tbinom ). However, it is not strongly regular when k>2, as different pairs of nonadjacent verti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Independent Set
In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a set S of vertices such that for every two vertices in S, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in S. A set is independent if and only if it is a clique in the graph's complement. The size of an independent set is the number of vertices it contains. Independent sets have also been called "internally stable sets", of which "stable set" is a shortening. A maximal independent set is an independent set that is not a proper subset of any other independent set. A maximum independent set is an independent set of largest possible size for a given graph G. This size is called the independence number of ''G'' and is usually denoted by \alpha(G). The optimization problem of finding such a set is called the maximum independent set problem. It is a strongly NP-hard problem. As such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle-free
In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs. By Turán's theorem, the ''n''-vertex triangle-free graph with the maximum number of edges is a complete bipartite graph in which the numbers of vertices on each side of the bipartition are as equal as possible. Triangle finding problem The triangle finding problem is the problem of determining whether a graph is triangle-free or not. When the graph does contain a triangle, algorithms are often required to output three vertices which form a triangle in the graph. It is possible to test whether a graph with edges is triangle-free in time . Another approach is to find the trace of , where is the adjacency matrix of the graph. The trace is zero if and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Spectrum
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. The adjacency matrix of a simple undirected graph is a real symmetric matrix and is therefore orthogonally diagonalizable; its eigenvalues are real algebraic integers. While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, although not a complete one. Spectral graph theory is also concerned with graph parameters that are defined via multiplicities of eigenvalues of matrices associated to the graph, such as the Colin de Verdière number. Cospectral graphs Two graphs are called cospectral or isospectral if the adjacency matrices of the graphs are isospectral, that is, if the adjacency matrices have equal multisets of eigenvalues. Cospectral graphs need not be isomorphic, but isomorphic graphs ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Kneser Graph")
