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Cage (graph Theory)
In the mathematics, mathematical field of graph theory, a cage is a regular graph that has as few vertex (graph theory), vertices as possible for its girth (graph theory), girth. Formally, an is defined to be a graph (discrete mathematics), graph in which each vertex has exactly neighbors, and in which the shortest cycle (graph theory), cycle has a length of exactly . An is an with the smallest possible number of vertices, among all . A is often called a . It is known that an exists for any combination of and . It follows that all exist. If a Moore graph exists with degree and girth , it must be a cage. Moreover, the bounds on the sizes of Moore graphs generalize to cages: any cage with parity (mathematics), odd girth must have at least :1 + r\sum_^(r-1)^i vertices, and any cage with parity (mathematics), even girth must have at least :2\sum_^(r-1)^i vertices. Any with exactly this many vertices is by definition a Moore graph and therefore automatically a cage. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tutte Eight Cage
William Thomas Tutte (; 14 May 1917 – 2 May 2002) was an English and Canadian Cryptanalysis, code breaker and mathematician. During the Second World War, he made a fundamental advance in cryptanalysis of the Lorenz cipher, a major Nazi Germany, Nazi German cipher system which was used for top-secret communications within the Wehrmacht High Command. The high-level, strategic nature of the intelligence obtained from Tutte's crucial breakthrough, in the bulk decrypting of Lorenz-enciphered messages specifically, contributed greatly, and perhaps even decisively, to the defeat of Nazi Germany. He also had a number of significant mathematical accomplishments, including foundation work in the fields of graph theory and matroid theory. Tutte's research in the field of graph theory proved to be of remarkable importance. At a time when graph theory was still a primitive subject, Tutte commenced the study of matroids and developed them into a theory by expanding from the work that Hass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Bipartite Graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher. Llull himself had made similar drawings of complete graphs three centuries earlier.. Definition A complete bipartite graph is a graph whose vertices can be partitioned into two subsets and such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph such that for every two vertices and, is an edge in . A complete bipartite graph w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan Graph
In the mathematical field of spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent expander graph, spectral expanders. As Murty's survey paper notes, Ramanujan graphs "fuse diverse branches of pure mathematics, namely, number theory, representation theory, and algebraic geometry". These graphs are indirectly named after Srinivasa Ramanujan; their name comes from the Ramanujan–Petersson conjecture, which was used in a construction of some of these graphs. Definition Let G be a connected d-regular graph with n vertices, and let \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n be the eigenvalues of the adjacency matrix of G (or the Spectral graph theory, spectrum of G). Because G is connected and d-regular, its eigenvalues satisfy d = \lambda_1 > \lambda_2 \geq \cdots \geq \lambda_n \geq -d . Define \lambda(G) = \max_, \lambda_i, = \max(, \lambda_2, ,\ldots, , \lamb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , then is the logarithm of to base , written , so . As a single-variable function, the logarithm to base is the inverse of exponentiation with base . The logarithm base is called the ''decimal'' or ''common'' logarithm and is commonly used in science and engineering. The ''natural'' logarithm has the number as its base; its use is widespread in mathematics and physics because of its very simple derivative. The ''binary'' logarithm uses base and is widely used in computer science, information theory, music theory, and photography. When the base is unambiguous from the context or irrelevant it is often omitted, and the logarithm is written . Logarithms were introduced by John Napier in 1614 as a means of simplifying calculation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Growth
Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now. In more technical language, its instantaneous rate of change (that is, the derivative) of a quantity with respect to an independent variable is proportional to the quantity itself. Often the independent variable is time. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth. Not all cases of growth at an always increasing rate are instances of exponential growth. For example the function f(x) = x^3 grows at an ever increasing rate, but is much slower than growing exponentially. For example, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Polygon
In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized ''n''-gons encompass as special cases projective planes (generalized triangles, ''n'' = 3) and generalized quadrangles (''n'' = 4). Many generalized polygons arise from groups of Lie type, but there are also exotic ones that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the ''Ruth Moufang, Moufang property'' have been completely classified by Tits and Weiss. Every generalized ''n''-gon with ''n'' even is also a near polygon. Definition A generalized ''2''-gon (or a digon) is an incidence structure with at least 2 points and 2 lines where each point is incident to each line. For ''n \geq 3'' a generalized ''n''-gon is an incidence structure (P,L,I), where P is the set of points, L is the set of lines and I\subseteq P\times L is the incidence relation, such that: * It is a partial linear space. * It has no ordinary ''m''- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in Perspective (graphical)#Renaissance, perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Power
In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, … . The prime powers are those positive integers that are divisible by exactly one prime number; in particular, the number 1 is not a prime power. Prime powers are also called primary numbers, as in the primary decomposition. Properties Algebraic properties Prime powers are powers of prime numbers. Every prime power (except powers of 2 greater than 4) has a primitive root; thus the multiplicative group of integers modulo ''p''''n'' (that is, the group of units of the ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hoffman–Singleton Graph
In the mathematical field of graph theory, the Hoffman–Singleton graph is a undirected graph with 50 vertices and 175 edges. It is the unique strongly regular graph with parameters (50,7,0,1). It was constructed by Alan Hoffman and Robert Singleton while trying to classify all Moore graphs, and is the highest-order Moore graph known to exist. Since it is a Moore graph where each vertex has degree 7, and the girth is 5, it is a . Construction Here are some constructions of the Hoffman–Singleton graph. Construction from pentagons and pentagrams Take five pentagons ''Ph'' and five pentagrams ''Qi'' . Join vertex ''j'' of ''Ph'' to vertex ''h'' · ''i'' + ''j'' of ''Qi'' (all indices are modulo 5.) Construction from PG(3,2) Take a Fano plane on seven elements, such as and apply all 2520 even permutations on the ''abcdefg''. Canonicalize each such Fano plane (e.g. by reducing to lexicographic order) and discard duplicates. Exactly 15 Fano planes remain. Each (tripl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robertson Graph
In the mathematical field of graph theory, the Robertson graph or (4,5)-cage, is a 4- regular undirected graph with 19 vertices and 38 edges named after Neil Robertson. The Robertson graph is the unique (4,5)-cage graph and was discovered by Robertson in 1964. As a cage graph, it is the smallest 4-regular graph with girth 5. It has chromatic number 3, chromatic index 5, diameter 3, radius 3 and is both 4- vertex-connected and 4- edge-connected. It has book thickness 3 and queue number 2.Jessica Wolz, ''Engineering Linear Layouts with SAT''. Master Thesis, University of Tübingen, 2018 The Robertson graph is also a Hamiltonian graph which possesses distinct directed Hamiltonian cycles. The Robertson graph is one of the smallest graphs with cop number 4.Turcotte, J., & Yvon, S. (2021). 4-cop-win graphs have at least 19 vertices. Discrete Applied Mathematics, 301, 74-98. Algebraic properties The Robertson graph is not a vertex-transitive graph; its full automorphism gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tutte 12-cage
In the mathematical field of graph theory, the Tutte 12-cage or Benson graph is a 3-regular graph with 126 vertices and 189 edges. It is named after W. T. Tutte. The Tutte 12-cage is the unique (3-12)- cage . It was discovered by C. T. Benson in 1966. It has chromatic number 2 ( bipartite), chromatic index 3, girth 12 (as a 12-cage) and diameter 6. Its crossing number is known to be less than 165see Wolfram MathWorld. Construction The Tutte 12-cage is a cubic Hamiltonian graph and can be defined by the LCF notation 7, 27, −13, −59, −35, 35, −11, 13, −53, 53, −27, 21, 57, 11, −21, −57, 59, −17sup>7. There are, up to isomorphism, precisely two generalized hexagons of order ''(2,2)'' as proved by Cohen and Tits. They are the split Cayley hexagon ''H(2)'' and its point-line dual. Clearly both of them have the same incidence graph, which is in fact isomorphic to the Tutte 12-cage. The Balaban 11-cage can be constructed by excision from the Tutte 12-cage by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tutte–Coxeter Graph
In the mathematics, mathematical field of graph theory, the Tutte–Coxeter graph or Tutte eight-cage or Cremona–Richmond graph is a 3-regular graph with 30 vertices and 45 edges. As the unique smallest cubic graph of girth (graph theory), girth 8, it is a cage (graph theory), cage and a Moore graph. It is bipartite graph, bipartite, and can be constructed as the Levi graph of the generalized quadrangle ''W''2 (known as the Cremona–Richmond configuration). The graph is named after William Thomas Tutte and H. S. M. Coxeter; it was discovered by Tutte (1947) but its connection to geometric configurations was investigated by both authors in a pair of jointly published papers (Tutte 1958; Coxeter 1958a). All the cubic graph, cubic distance-regular graphs are known. The Tutte–Coxeter is one of the 13 such graphs. It has Crossing number (graph theory), crossing number 13, book thickness 3 and queue number 2.Wolz, Jessica; ''Engineering Linear Layouts with SAT.'' Master Thesis, Un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
