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Conway's 99-graph Problem
In graph theory, Conway's 99-graph problem is an unsolved problem asking whether there exists an undirected graph with 99 vertices, in which each two adjacent vertices have exactly one common neighbor, and in which each two non-adjacent vertices have exactly two common neighbors. Equivalently, every edge should be part of a unique triangle and every non-adjacent pair should be one of the two diagonals of a unique 4-cycle. John Horton Conway offered a $1000 prize for its solution. Properties If such a graph exists, it would necessarily be a locally linear graph and a strongly regular graph with parameters (99,14,1,2). The first, third, and fourth parameters encode the statement of the problem: the graph should have 99 vertices, every pair of adjacent vertices should have 1 common neighbor, and every pair of non-adjacent vertices should have 2 common neighbors. The second parameter means that the graph is a regular graph with 14 edges per vertex. If this graph exists, it cannot ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Regular Graph
In graph theory, a strongly regular graph (SRG) is a regular graph with vertices and degree such that for some given integers \lambda, \mu \ge 0 * every two adjacent vertices have common neighbours, and * every two non-adjacent vertices have common neighbours. Such a strongly regular graph is denoted by . Its complement graph is also strongly regular: it is an . 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 as 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 bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Danzer Set
In geometry, a Danzer set is a set of points that touches every convex body of unit volume. Ludwig Danzer asked whether it is possible for such a set to have bounded density. Several variations of this problem remain unsolved. Formulation A ''Danzer set'', in an -dimensional Euclidean space, is a set of points in the space that has a non-empty intersection with every convex body whose -dimensional volume is one. The whole space is itself a Danzer set, but it is possible for a Danzer set to be a discrete set with only finitely many points in any bounded area. Danzer's question asked whether, more strongly, the average number of points per unit area could be bounded. One way to define the problem more formally is to consider the growth rate of a set S in Euclidean space, defined as the function that maps a real number r to the number of points of S that are within distance r of the origin. Danzer's question is whether it is possible for a Danzer set to have growth expressed in b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Regular Graphs
In graph theory, a strongly regular graph (SRG) is a regular graph with vertices and Degree (graph theory), degree such that for some given integers \lambda, \mu \ge 0 * every two adjacent vertices have common neighbours, and * every two non-adjacent vertices have common neighbours. Such a strongly regular graph is denoted by . Its complement graph is also strongly regular: it is an . 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 as 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 complement graph, complements, the complete multipartite graphs with equal-sized independent sets. Andries Brouwer and Hendrik van Maldeghem (see ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
On-Line Encyclopedia Of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the OEIS Foundation in 2009, and is its chairman. OEIS records information on integer sequences of interest to both professional and amateur mathematicians, and is widely cited. , it contains over 370,000 sequences, and is growing by approximately 30 entries per day. Each entry contains the leading terms of the sequence, keywords, mathematical motivations, literature links, and more, including the option to generate a graph or play a musical representation of the sequence. The database is searchable by keyword, by subsequence, or by any of 16 fields. There is also an advanced search function called SuperSeeker which runs a large number of different algorithms to identify sequences related to the input. History Neil Sloane started col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Mathematics (journal)
''Discrete Mathematics'' is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications. It was established in 1971 and is published by North-Holland Publishing Company. It publishes both short notes, full length contributions, as well as survey articles. In addition, the journal publishes a number of special issues each year dedicated to a particular topic. Although originally it published articles in French and German, it now allows only English language articles. The editor-in-chief is Douglas West ( University of Illinois, Urbana). History The journal was established in 1971. The first article it published was written by Paul Erdős, who went on to publish a total of 84 papers in the journal. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact facto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-3 Duoprism
In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. Descriptions The duoprism is a 4-polytope that can be constructed using Cartesian product of two polygons. In the case of 3-3 duoprism is the simplest among them, and it can be constructed using Cartesian product of two triangles. The resulting duoprism has 9 vertices, 18 edges, and 15 faces—which include 9 squares and 6 triangles. Its cell has 6 triangular prism. It has Coxeter diagram , and symmetry , order 72. The hypervolume of a uniform 3-3 duoprism with edge length a is V_4 = a^4. This is the square of the area of an equilateral triangle, A = a^2. The 3-3 duoprism can be represented as a graph with the same number of vertices and edges. Like the Berlekamp–van Lint–Seidel graph and the unknown solution to Conway's 99-graph problem, every edge is part of a unique triangle and every non-adjacent pair of vertices is the diagonal of a unique squar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paley Graph
In mathematics, Paley graphs are undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrix, conference matrices. Paley graphs allow graph-theoretic tools to be applied to the number theory of quadratic residues, and have interesting properties that make them useful in graph theory more generally. Paley graphs are named after Raymond Paley. They are closely related to the Paley construction for constructing Hadamard matrix, Hadamard matrices from quadratic residues. They were introduced as graphs independently by and . Horst Sachs, Sachs was interested in them for their self-complementarity properties, while Paul Erdős, Erdős and Alfréd Rényi, Rényi studied their symmetries. Paley digraphs are directed graph, directed analogs of Paley graphs that yield antisymmetric conf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylver Coinage
Sylver coinage is a mathematical game for two players, invented by John H. Conway. The two players take turns naming positive integers that are not the sum of nonnegative multiples of previously named integers. The player who names 1 loses. For instance, if player A opens with 2, B can win by naming 3 as A is forced to name 1. Sylver coinage is an example of a game using misère play because the player who is last able to move loses. Sylver coinage is named after James Joseph Sylvester, who proved that if ''a'' and ''b'' are relatively prime positive integers, then (''a'' − 1)(''b'' − 1) − 1 is the largest number that is not a sum of nonnegative multiples of ''a'' and ''b''. Thus, if ''a'' and ''b'' are the first two moves in a game of sylver coinage, this formula gives the largest number that can still be played. More generally, if the greatest common divisor of the moves played so far is ''g'', then only finitely many multiples of ''g'' can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thrackle
A thrackle is an embedding of a graph in the plane in which each edge is a Jordan arc and every pair of edges meet exactly once. Edges may either meet at a common endpoint, or, if they have no endpoints in common, at a point in their interiors. In the latter case, they must cross at their intersection point: the intersection must be ''transverse''.. A preliminary version of these results was reviewed in . A special case of thrackles, the linear thrackles, restrict the edges to be drawn as straight line segments. One method for constructing a linear thrackle with any given set of points as vertices is to form an edge between each farthest pair of points. For a linear thrackle, each connected component contains at most one cycle, from which it follows that the number of edges is at most equal to the number of vertices. John H. Conway conjectured more generally that every thrackle has at most as many edges as vertices. It is known that the number of edges is at most a constant tim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman L
Norman or Normans may refer to: Ethnic and cultural identity * The Normans, a people partly descended from Norse Vikings who settled in the territory of Normandy in France in the 9th and 10th centuries ** People or things connected with the Norman conquest of southern Italy in the 11th and 12th centuries ** Normanist theory (also known as Normanism) and anti-Normanism, historical disagreement regarding the origin of Russia, Ukraine, Belarus and their historic predecessor, Kievan Rus' ** Norman dynasty, a series of monarchs in England and Normandy ** Norman architecture, romanesque architecture in England and elsewhere ** Norman language, spoken in Normandy ** People or things connected with the French region of Normandy Arts and entertainment * ''Norman'' (2010 film), a 2010 drama film * ''Norman'' (2016 film), a 2016 drama film * ''Norman'' (TV series), a 1970 British sitcom starring Norman Wisdom * ''The Normans'' (TV series), a documentary * "Norman" (song), a 1962 song w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
