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Geodetic Graph
In graph theory, a geodetic graph is an undirected graph such that there exists a unique (unweighted) shortest path between each two vertices. Geodetic graphs were introduced in 1962 by Øystein Ore, who observed that they generalize a property of trees (in which there exists a unique path between each two vertices regardless of distance), and asked for a characterization of them. Although these graphs can be recognized in polynomial time, "more than sixty years later a full characterization is still elusive". Examples Every tree, every complete graph, and every odd-length cycle graph is geodetic. If G is a geodetic graph, then replacing every edge of G by a path of the same odd length will produce another geodetic graph. In the case of a complete graph, a more general pattern of replacement by paths is possible: choose a non-negative integer f(v) for each vertex v, and subdivide each edge uv by adding f(u)+f(v) vertices to it. Then the resulting subdivided complete graph is geod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Undirected Graph
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diamond Graph
In the mathematical field of graph theory, the diamond graph is a planar, undirected graph with 4 vertices and 5 edges. It consists of a complete graph minus one edge. The diamond graph has radius 1, diameter 2, girth 3, chromatic number 3 and chromatic index 3. It is also a 2- vertex-connected and a 2- edge-connected, graceful, Hamiltonian graph. Diamond-free graphs and forbidden minor A graph is diamond-free if it has no diamond as an induced subgraph. The triangle-free graphs are diamond-free graphs, since every diamond contains a triangle. The diamond-free graphs are locally clustered: that is, they are the graphs in which every neighborhood is a cluster graph. Alternatively, a graph is diamond-free if and only if every pair of maximal cliques in the graph shares at most one vertex. The family of graphs in which each connected component is a cactus graph is downwardly closed under graph minor operations. This graph family may be characterized by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
European Journal Of Combinatorics
European, or Europeans, or Europeneans, may refer to: In general * ''European'', an adjective referring to something of, from, or related to Europe ** Ethnic groups in Europe ** Demographics of Europe ** European cuisine, the cuisines of Europe and other Western countries * ''European'', an adjective referring to something of, from, or related to the European Union ** Citizenship of the European Union ** Demographics of the European Union In publishing * ''The European'' (1953 magazine), a far-right cultural and political magazine published 1953–1959 * ''The European'' (newspaper), a British weekly newspaper published 1990–1998 * ''The European'' (2009 magazine), a German magazine first published in September 2009 *''The European Magazine'', a magazine published in London 1782–1826 *''The New European'', a British weekly pop-up newspaper first published in July 2016 Other uses * * Europeans (band), a British post-punk group, from Bristol See also * * * Europe (disam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doklady Akademii Nauk
The ''Proceedings of the USSR Academy of Sciences'' (russian: Доклады Академии Наук СССР, ''Doklady Akademii Nauk SSSR'' (''DAN SSSR''), french: Comptes Rendus de l'Académie des Sciences de l'URSS) was a Soviet journal that was dedicated to publishing original, academic research papers in physics, mathematics, chemistry, geology, and biology. It was first published in 1933 and ended in 1992 with volume 322, issue 3. Today, it is continued by ''Doklady Akademii Nauk'' (russian: Доклады Академии Наук), which began publication in 1992. The journal is also known as the ''Proceedings of the Russian Academy of Sciences (RAS)''. ''Doklady'' has had a complicated publication and translation history. A number of translation journals exist which publish selected articles from the original by subject section; these are listed below. History The Russian Academy of Sciences dates from 1724, with a continuous series of variously named publications dat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometriae Dedicata
''Geometriae Dedicata'' is a mathematical journal, founded in 1972, concentrating on geometry and its relationship to topology, group theory and the theory of dynamical systems. It was created on the initiative of Hans Freudenthal in Utrecht, the Netherlands.. It is published by Springer Netherlands. The Editors-in-Chief An editor-in-chief (EIC), also known as lead editor or chief editor, is a publication's editorial leader who has final responsibility for its operations and policies. The highest-ranking editor of a publication may also be titled editor, managing ... are John R. Parker and Jean-Marc Schlenker.Journal website References External links Springer site Mathematics journals Springer Science+Business Media academic journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi Graph
In combinatorial mathematics, a Levi graph or incidence graph is a bipartite graph associated with an incidence structure.. See in particulap. 181 From a collection of points and lines in an incidence geometry or a projective configuration, we form a graph with one vertex per point, one vertex per line, and an edge for every incidence between a point and a line. They are named for Friedrich Wilhelm Levi, who wrote about them in 1942. The Levi graph of a system of points and lines usually has girth at least six: Any 4- cycles would correspond to two lines through the same two points. Conversely any bipartite graph with girth at least six can be viewed as the Levi graph of an abstract incidence structure. Levi graphs of configurations are biregular, and every biregular graph with girth at least six can be viewed as the Levi graph of an abstract configuration.. Levi graphs may also be defined for other types of incidence structure, such as the incidences between points and planes in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Plane (incidence Geometry)
In geometry, an affine plane is a system of points and lines that satisfy the following axioms: * Any two distinct points lie on a unique line. * Given any line and any point not on that line there is a unique line which contains the point and does not meet the given line. (Playfair's axiom) * There exist three non-collinear points (points not on a single line). In an affine plane, two lines are called ''parallel'' if they are equal or disjoint. Using this definition, Playfair's axiom above can be replaced by: * Given a point and a line, there is a unique line which contains the point and is parallel to the line. Parallelism is an equivalence relation on the lines of an affine plane. Since no concepts other than those involving the relationship between points and lines are involved in the axioms, an affine plane is an object of study belonging to incidence geometry. They are non-degenerate linear spaces satisfying Playfair's axiom. The familiar Euclidean plane is an affine pla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hoffman–Singleton Graph
In the mathematical field of graph theory, the Hoffman–Singleton graph is a 7- regular 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 (7,5)-cage. Construction Here are two 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 7-set ''abcdefg''. Canonicalize each such Fano plane (e.g. by reducing to lexicographic order) and discard duplicates. Exactly 15 Fano planes remai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petersen Graph
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with no three-edge-coloring. Although the graph is generally credited to Petersen, it had in fact first appeared 12 years earlier, in a paper by . Kempe observed that its vertices can represent the ten lines of the Desargues configuration, and its edges represent pairs of lines that do not meet at one of the ten points of the configuration. Donald Knuth states that the Petersen graph is "a remarkable configuration that serves as a counterexample to many optimistic predictions about what might be true for graphs in general." The Petersen graph also makes an appearance in tropical geometry. The cone over the Petersen graph is naturally identif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G, denoted by \Delta(G), and the minimum degree of a graph, denoted by \delta(G), are the maximum and minimum of its vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is entitled negative deg(v). Handshaking lemma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Regular Graph
In graph theory, a strongly regular graph (SRG) is defined as follows. Let be a regular graph with vertices and degree . is said to be strongly regular if there are also integers and such that: * Every two adjacent vertices have common neighbours. * Every two non-adjacent vertices have common neighbours. The complement of an is also strongly regular. It is a . 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 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 but fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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