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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 It was discovered by Marion C. Gray in 1932 (unpublished), then discovered independently by Bouwer 1968 in reply to a question posed by Jon Folkman 1967.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] |
Gray Graph Hamiltonian
Grey (more frequent in British English) or gray (more frequent in American English) is an intermediate color between black and white. It is a neutral or achromatic color, meaning that it has no chroma. It is the color of a cloud-covered sky, of ash, and of lead. The first recorded use of ''grey'' as a color name in the English language was in 700 CE.Maerz and Paul ''A Dictionary of Color'' New York:1930 McGraw-Hill Page 196 ''Grey'' is the dominant spelling in European and Commonwealth English, while ''gray'' is more common in American English; however, both spellings are valid in both varieties of English. In Europe and North America, surveys show that gray is the color most commonly associated with neutrality, conformity, boredom, uncertainty, old age, indifference, and modesty. Only one percent of respondents chose it as their favorite color. Etymology ''Grey'' comes from the Middle English or , from the Old English , and is related to the Dutch and German . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-edge-connected Graph
In graph theory, a connected graph is -edge-connected if it remains connected whenever fewer than edges are removed. The edge-connectivity of a graph is the largest for which the graph is -edge-connected. Edge connectivity and the enumeration of -edge-connected graphs was studied by Camille Jordan in 1869. Formal definition Let G = (V, E) be an arbitrary graph. If the subgraph G' = (V, E \setminus X) is connected for all X \subseteq E where , X, < k, then ''G'' is said to be ''k''-edge-connected. The edge connectivity of is the maximum value ''k'' such that ''G'' is ''k''-edge-connected. The smallest set ''X'' whose removal disconnects ''G'' is a minimum cut in ''G''. The edge connectivity version of Menger's theorem provides an alternative and equivalent character ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. * '''' * '' |
Unit Distance Graph
In mathematics, particularly geometric graph theory, a unit distance graph is a Graph (discrete mathematics), graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exactly one. To distinguish these graphs from a broader definition that allows some non-adjacent pairs of vertices to be at distance one, they may also be called strict unit distance graphs or faithful unit distance graphs. As a hereditary property, hereditary family of graphs, they can be characterized by forbidden graph characterization, forbidden induced subgraphs. The unit distance graphs include the cactus graphs, the matchstick graphs and penny graphs, and the hypercube graphs. The generalized Petersen graphs are non-strict unit distance graphs. An unsolved problem of Paul Erdős asks how many edges a unit distance graph on n vertices can have. The best known lower bound is slightly above linear in n—far from the upper bound, proportional t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Graph Automorphism
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge– vertex connectivity. Formally, an automorphism of a graph is a permutation of the vertex set , such that the pair of vertices form an edge if and only if the pair also form an edge. That is, it is a graph isomorphism from to itself. Automorphisms may be defined in this way both for directed graphs and for undirected graphs. The composition of two automorphisms is another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphism group of the graph. In the opposite direction, by Frucht's theorem, all groups can be represented as the automorphism group of a connected graph – indeed, of a cubic graph. Computational complexity Constructing the automorphism group of a graph, in the form of a list of generators, is polynomial-time equivalent to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism Group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the group of invertible linear transformations from ''X'' to itself (the general linear group of ''X''). If instead ''X'' is a group, then its automorphism group \operatorname(X) is the group consisting of all group automorphisms of ''X''. Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups are studied in a general way in the field of category theory. Examples If ''X'' is a set with no additional structure, then any bijection from ''X'' to itself is an automorphism, and hence the automorphism group of ''X'' in this case is precisely the symmetric group of ''X''. If the set ''X'' has additional structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
LCF Notation
In the mathematical field of graph theory, LCF notation or LCF code is a notation devised by Joshua Lederberg, and extended by Harold Scott MacDonald Coxeter, H. S. M. Coxeter and Robert Frucht, for the representation of cubic graphs that contain a Hamiltonian path, Hamiltonian cycle. The cycle itself includes two out of the three adjacencies for each Vertex (graph theory), vertex, and the LCF notation specifies how far along the cycle each vertex's third neighbor is. A single graph may have multiple different representations in LCF notation. Description In a Hamiltonian graph, the vertices can be circular layout, arranged in a cycle, which accounts for two edges per vertex. The third edge from each vertex can then be described by how many positions clockwise (positive) or counter-clockwise (negative) it leads. The basic form of the LCF notation is just the sequence of these numbers of positions, starting from an arbitrarily chosen vertex and written in square brackets. The numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Girth (graph Theory)
In graph theory, the girth of an undirected graph is the length of a shortest Cycle (graph theory), cycle contained in the graph. If the graph does not contain any cycles (that is, it is a forest (graph theory), forest), its girth is defined to be infinity. For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3. A graph with girth four or more is triangle-free graph, triangle-free. Cages A cubic graph (all vertices have degree three) of girth that is as small as possible is known as a -cage (graph theory), cage (or as a -cage). The Petersen graph is the unique 5-cage (it is the smallest cubic graph of girth 5), the Heawood graph is the unique 6-cage, the McGee graph is the unique 7-cage and the Tutte eight cage is the unique 8-cage. There may exist multiple cages for a given girth. For instance there are three nonisomorphic 10-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries–Wong graph. Im ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle (graph Theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an ''acyclic graph''. A directed graph without directed cycles is called a '' directed acyclic graph''. A connected graph without cycles is called a ''tree''. Definitions Circuit and cycle * A circuit is a non-empty trail in which the first and last vertices are equal (''closed trail''). : Let be a graph. A circuit is a non-empty trail with a vertex sequence . * A cycle or simple circuit is a circuit in which only the first and last vertices are equal. * ''n'' is called the length of the circuit resp. length of the cycle. Directed circuit and directed cycle * A directed circuit is a non-empty directed trail in which the first and last vertices are equal (''closed directed trail''). : Let be a directed grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tomaž Pisanski
Tomaž (Tomo) Pisanski (born 24 May 1949 in Ljubljana, Yugoslavia, which is now in Slovenia) is a Slovenian mathematician working mainly in discrete mathematics and graph theory. He is considered by many Slovenian mathematicians to be the "father of Slovenian discrete mathematics." Biography As a high school student, Pisanski competed in the 1966 and 1967 International Mathematical Olympiads as a member of the Yugoslav team, winning a bronze medal in 1967. He studied at the University of Ljubljana where he obtained a B.Sc, M.Sc and PhD in mathematics. His 1981 PhD thesis in topological graph theory was written under the guidance of Torrence Parsons. He also obtained an M.Sc. in computer science from Pennsylvania State University in 1979. Currently, Pisanski is a professor of discrete and computational mathematics and Head of the Department of Information Sciences and Technology at University of Primorska in Koper. In addition, he is a professor at the University of Ljubljana Fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
