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Odd Graph
In the mathematics, mathematical field of graph theory, the odd graphs are a family of symmetric graphs defined from certain set systems. They include and generalize the Petersen graph. The odd graphs have high odd girth, meaning that they contain long parity (mathematics), odd-length cycle (graph theory), cycles but no short ones. However their name comes not from this property, but from the fact that each edge (graph theory), edge in the graph has an "odd man out", an element that does not participate in the two sets connected by the edge. Definition and examples The odd graph O_n has one vertex (graph theory), vertex for each of the (n-1)-element subsets of a (2n-1)-element set. Two vertices are connected by an edge if and only if the corresponding subsets are disjoint sets, disjoint. That is, O_n is the Kneser graph KG(2n-1,n-1). O_2 is a triangle, while O_3 is the familiar Petersen graph. The generalized odd graphs are defined as distance-regular graphs with diameter (grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kneser Graph KG(5,2)
Kneser is a surname. Notable people with the surname include: *Adolf Kneser (1862–1930), mathematician *Hellmuth Kneser Hellmuth Kneser (16 April 1898 – 23 August 1973) was a German mathematician who made notable contributions to group theory and topology. His most famous result may be his theorem on the existence of a prime decomposition for 3-manifolds. His ... (1898–1973), mathematician, son of Adolf Kneser * Martin Kneser (1928–2004), mathematician, son of Hellmuth Kneser {{surname ... [...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 . Some authors exclude the complete graphs and disconnected graphs from this definition. 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 The intersection array of a distance-regular graph is the array ( b_0, b_1, \ldots, b_; c_1, \ldots, c_d ) in which d is the diameter of the graph and for each 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 dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Graph
In the mathematical field of graph theory, a graph is symmetric or arc-transitive if, given any two ordered pairs of adjacent vertices (u_1,v_1) and (u_2,v_2) 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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shortest Path
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length or distance of each segment. Definition The shortest path problem can be defined for graphs whether undirected, directed, or mixed. The definition for undirected graphs states that every edge can be traversed in either direction. Directed graphs require that consecutive vertices be connected by an appropriate directed edge. Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices P = ( v_1, v_2, \ldots, v_n ) \in V \times V \times \cdots \times V suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Graph
In graph theory, a regular graph is a Graph (discrete mathematics), graph where each Vertex (graph theory), vertex has the same number of neighbors; i.e. every vertex has the same Degree (graph theory), degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each internal vertex are equal to each other. A regular graph with vertices of degree is called a graph or regular graph of degree . Special cases Regular graphs of degree at most 2 are easy to classify: a graph consists of disconnected vertices, a graph consists of disconnected edges, and a graph consists of a disjoint union of graphs, disjoint union of cycle (graph theory), cycles and infinite chains. A graph is known as a cubic graph. A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number of neighbors in common, and every non-adjacent pair of vertices has the same number of neighbors in common. The smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Association Of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university A university () is an educational institution, institution of tertiary education and research which awards academic degrees in several Discipline (academia), academic disciplines. ''University'' is derived from the Latin phrase , which roughly ..., college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists; statisticians; and many others in academia, government, business, and industry. The MAA was founded in 1915 and is headquartered at 11 Dupont in the Dupont Circle, Washington, D.C., Dupont Circle neighborhood of Washington, D.C. The organization publishes mathematics journals and books, including the ''American Mathematical Monthly'' (established in 1894 by Benjamin Finkel), the most widely read mathematics journal in the world according to re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odd One Out
''Odd One Out'' is a British game show based on the American version entitled ''Knockout''. It aired on BBC1 from 16 April 1982 to 19 April 1985 and was hosted by Paul Daniels. The show is based on a short-lived American game show produced by Ralph Edwards called ''Knockout'', hosted by Arte Johnson Arthur Stanton Eric Johnson (January 20, 1929 – July 3, 2019) was an American actor and comedian who was best known for his work as a regular on television's ''Rowan & Martin's Laugh-In''. Biography Early life Johnson was born January 20, 19 .... Gameplay The object of ''Odd One Out'' is to guess which one of four items does not belong and why it doesn't belong. After the player has successfully identified the odd one out, they can either guess the explanation or challenge their opponents to guess. Choosing the correct item would score two points, and figuring out why it didn't belong by guessing the common bond of the other three or a successful challenge is worth three poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tony Gardiner
Tony Gardiner (17 May 1947 – 22 January 2024) was a British mathematician who until 2012 held the position of Reader in Mathematics and Mathematics Education at the University of Birmingham. He was responsible for the foundation of the United Kingdom Mathematics Trust in 1996, one of the UK's largest mathematics enrichment programs, initiating the Intermediate and Junior Mathematical Challenges, creating the Problem Solving Journal for secondary school students and organising numerous masterclasses, summer schools and educational conferences. Gardiner contributed to many educational articles and internationally circulated educational pamphlets. As well as his involvement with mathematics education, Gardiner has also made contributions to the areas of infinite groups, finite groups, graph theory, and algebraic combinatorics. At the time of his death he was still a member of UKMT. In the year 1994–1995, he received the Paul Erdős Award for his contributions to UK and internat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman Biggs (mathematician)
Norman Linstead Biggs (born 2 January 1941) is a leading British mathematician focusing on discrete mathematics and in particular algebraic combinatorics.. Education Biggs was educated at Harrow County Grammar School and then studied mathematics at Selwyn College, Cambridge. In 1962, Biggs gained first-class honours in his third year of the university's undergraduate degree in mathematics. *1946–1952: Uxendon Manor Primary School, Kenton, Middlesex *1952–1959: Harrow County Grammar School *1959–1963: Selwyn College, Cambridge (Entrance Exhibition 1959, Scholarship 1961) *1960: First Class, Mathematical Tripos Pt. I *1962: Wrangler, Mathematical Tripos Pt. II; B.A. (Cantab.) *1963: Distinction, Mathematical Tripos Pt. III *1988: D.Sc. (London); M.A. (Cantab.) Career He was a lecturer at University of Southampton, lecturer then reader at Royal Holloway, University of London, and Professor of Mathematics at the London School of Economics. He has been on the editorial board ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Computing
Parallel computing is a type of computing, computation in which many calculations or Process (computing), processes are carried out simultaneously. Large problems can often be divided into smaller ones, which can then be solved at the same time. There are several different forms of parallel computing: Bit-level parallelism, bit-level, Instruction-level parallelism, instruction-level, Data parallelism, data, and task parallelism. Parallelism has long been employed in high-performance computing, but has gained broader interest due to the physical constraints preventing frequency scaling.S.V. Adve ''et al.'' (November 2008)"Parallel Computing Research at Illinois: The UPCRC Agenda" (PDF). Parallel@Illinois, University of Illinois at Urbana-Champaign. "The main techniques for these performance benefits—increased clock frequency and smarter but increasingly complex architectures—are now hitting the so-called power wall. The computer industry has accepted that future performance inc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Network Topology
Network topology is the arrangement of the elements (Data link, links, Node (networking), nodes, etc.) of a communication network. Network topology can be used to define or describe the arrangement of various types of telecommunication networks, including command and control radio networks, industrial Fieldbus, fieldbusses and computer networks. Network topology is the topological structure of a network and may be depicted physically or logically. It is an application of graph theory wherein communicating devices are modeled as nodes and the connections between the devices are modeled as links or lines between the nodes. Physical topology is the placement of the various components of a network (e.g., device location and cable installation), while logical topology illustrates how data flows within a network. Distances between nodes, physical interconnections, transmission rates, or signal types may differ between two different networks, yet their logical topologies may be identica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carbonium Ion
In chemistry, a carbonium ion is a cation that has a pentacoordinated carbon atom. They are a type of carbocation. In older literature, the name "carbonium ion" was used for what is today called carbenium. Carbonium ions charge is delocalized in three-center, two-electron bonds. The more stable members are often bi- or polycyclic. 2-Norbornyl cation The 2-norbornyl cation is one of the best-characterized carbonium ions. It is the prototype for non-classical ions. As indicated first by low-temperature NMR spectroscopy and confirmed by X-ray crystallography, it has a symmetric structure with an RCH2+ group bonded to an alkene group, stabilized by a bicyclic structure. Cyclopropylmethyl cation A non-classical structure for is supported by substantial experimental evidence from solvolysis experiments and NMR studies. One or both of two structures, the cyclopropylcarbinyl cation and the bicyclobutonium cation, were invoked to account for the observed reactivity. The NMR spectr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
