|
Semi-symmetric Graph
In the mathematical field of graph theory, a semi-symmetric graph is an undirected graph that is edge-transitive and regular, but not vertex-transitive. In other words, a graph is semi-symmetric if each vertex has the same number of incident edges, and there is a symmetry taking any of the graph's edges to any other of its edges, but there is some pair of vertices such that no symmetry maps the first into the second. Properties A semi-symmetric graph must be bipartite, and its automorphism group must act transitively on each of the two vertex sets of the bipartition (in fact, regularity is not required for this property to hold). For instance, in the diagram of the Folkman graph shown here, green vertices can not be mapped to red ones by any automorphism, but every two vertices of the same color are symmetric with each other. History Semi-symmetric graphs were first studied E. Dauber, a student of F. Harary, in a paper, no longer available, titled "On line- but not point-sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Folkman Graph
In the mathematical field of graph theory, the Folkman graph is a 4- regular graph with 20 vertices and 40 edges. It is a regular bipartite graph with symmetries taking every edge to every other edge, but the two sides of its bipartition are not symmetric with each other, making it the smallest possible semi-symmetric graph. It is named after Jon Folkman, who constructed it for this property in 1967. The Folkman graph can be constructed either using modular arithmetic or as the subdivided double of the five-vertex complete graph. Beyond the investigation of its symmetry, it has also been investigated as a counterexample for certain questions of graph embedding. Construction Semi-symmetric graphs are defined as regular graphs (that is, graphs in which all vertices touch equally many edges) in which each two edges are symmetric to each other, but some two vertices are not symmetric. Jon Folkman was inspired to define and research these graphs in a 1967 paper, after seeing an unp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Graph
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bipartite graph. Symmetry In 1932, Ronald M. Foster began collecting examples of cubic symmetric graphs, forming the start of the Foster census.. Many well-known individual graphs are cubic and symmetric, including the utility graph, the Petersen graph, the Heawood graph, the Möbius–Kantor graph, the Pappus graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs–Smith graph. W. T. Tutte classified the symmetric cubic graphs by the smallest integer number ''s'' such that each two oriented paths of length ''s'' can be mapped to each other by exactly one symmetry of the graph. He showed that ''s'' is at most 5, and provided examples of graphs with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Graph Theory
Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph invariants. Branches of algebraic graph theory Using linear algebra The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra. Especially, it studies the spectrum of the adjacency matrix, or the Laplacian matrix of a graph (this part of algebraic graph theory is also called spectral graph theory). For the Petersen graph, for example, the spectrum of the adjacency matrix is (−2, −2, −2, −2, 1, 1, 1, 1, 1, 3). Several theorems relate properties of the spectrum to other graph properties. As a simple example, a connected graph with diameter ''D'' wil ... [...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] |
Ljubljana Graph
In the mathematical field of graph theory, the Ljubljana graph is an undirected bipartite graph with 112 vertices and 168 edges, rediscovered in 2002 and named after Ljubljana (the capital of Slovenia). Conder, M.; Malnič, A.; Marušič, D.; Pisanski, T.; and Potočnik, P. "The Ljubljana Graph." 2002 It is a cubic graph with diameter 8, radius 7, chromatic number 2 and chromatic index 3. Its girth is 10 and there are exactly 168 cycles of length 10 in it. There are also 168 cycles of length 12. Construction The Ljubljana graph is Hamiltonian and can be constructed from the LCF notation : [47, −23, −31, 39, 25, −21, −31, −41, 25, 15, 29, −41, −19, 15, −49, 33, 39, −35, −21, 17, −33, 49, 41, 31, −15, −29, 41, 31, −15, −25, 21, 31, −51, −25, 23, 9, −17, 51, 35, −29, 21, −51, −39, 33, −9, −51, 51, −47, −33, 19, 51, −21, 29, 21, −31, −39]2. The Ljubljana graph is the Levi graph of the Ljubljana configuration, a quadrangle-fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marston Conder
Marston Donald Edward Conder (born 9 September 1955) is a New Zealand mathematician, a Distinguished Professor of Mathematics at Auckland University,Staff directory listing entry Auckland U. Mathematics, retrieved 22 January 2013. and the former co-director of the New Zealand Institute of Mathematics and its Applications. His main research interests are in , , and their connections with each other. Education and career Conder was born in |
Canadian Mathematical Bulletin
The ''Canadian Mathematical Bulletin'' () is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current editors-in-chief of the journal are Antonio Lei and Javad Mashreghi. The journal publishes short articles in all areas of mathematics that are of sufficient interest to the general mathematical public. Abstracting and indexing The journal is abstracted in: for the Canadian Mathematical Bulletin. * '''' * '' |
Dragan Marušič
Dragan Marušič (born 1953, Koper, Slovenia) is a Slovene mathematician. Marušič obtained his BSc in technical mathematics from the University of Ljubljana in 1976, and his PhD from the University of Reading in 1981 under the supervision of Crispin Nash-Williams. Marušič has published extensively, and has supervised seven PhD students (as of 2013). He served as the third rector of the University of Primorska from 2011 to 2019, a university he lobbied to have established in his home town of Koper. His research focuses on topics in algebraic graph theory, particularly the symmetry of graphs and the action of finite groups on combinatorial objects. He is regarded as the founder of the Slovenian school of research in algebraic graph theory and permutation groups. Education and career From 1968 to 1972 Marušič attended gymnasium in Koper. He studied undergraduate mathematics at the University of Ljubljana, graduating in 1976. He completed his PhD in 1981 in England, at the U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gray Graph
In the mathematical field of graph theory, the Gray graph is an undirected bipartite graph with 54 vertices and 81 edges. It is a cubic graph: every vertex touches exactly three edges. It was discovered by Marion C. Gray in 1932 (unpublished), then discovered independently by Bouwer 1968 in reply to a question posed by Jon Folkman 1967. The Gray graph is interesting as the first known example of a cubic graph having the algebraic property of being edge but not vertex transitive (see below). The Gray graph has chromatic number 2, chromatic index 3, radius 6 and diameter 6. It is also a 3- vertex-connected and 3- edge-connected non-planar graph. Construction The Gray graph can be constructed from the 27 points of a 3 × 3 × 3 grid and the 27 axis-parallel lines through these points. This collection of points and lines forms a projective configuration: each point has exactly three lines through it, and each line has exactly three points on it. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jon Folkman
Jon Hal Folkman (December 8, 1938 – January 23, 1969) was an American mathematician, a student of John Milnor, and a researcher at the RAND Corporation. Schooling Folkman was a William Lowell Putnam Mathematical Competition, Putnam Fellow in 1960. He received his Ph.D. in 1964 from Princeton University, under the supervision of Milnor, with a thesis entitled ''Equivariant Maps of Spheres into the Classical Groups''. Research Jon Folkman contributed important theorems in many areas of combinatorics. In geometric combinatorics, Folkman is known for his pioneering and posthumously-published studies of oriented matroids; in particular, the Folkman–Lawrence topological representation theorem is "one of the cornerstones of the theory of oriented matroids". In lattice (order), lattice theory, Folkman solved an open problem on the foundations of enumerative combinatorics, combinatorics by proving a conjecture of Gian-Carlo Rota, Gian–Carlo Rota; in proving Rota's conjecture ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transitive Group Action
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
