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Desargues Graph
In the mathematical field of graph theory, the Desargues graph is a distance-transitive, cubic graph with 20 vertices and 30 edges. It is named after Girard Desargues, arises from several different combinatorial constructions, has a high level of symmetry, is the only known non-planar cubic partial cube, and has been applied in chemical databases. The name "Desargues graph" has also been used to refer to a ten-vertex graph, the complement of the Petersen graph, which can also be formed as the bipartite half of the 20-vertex Desargues graph. Constructions There are several different ways of constructing the Desargues graph: *It is the generalized Petersen graph . To form the Desargues graph in this way, connect ten of the vertices into a regular decagon, and connect the other ten vertices into a ten-pointed star that connects pairs of vertices at distance three in a second decagon. The Desargues graph consists of the 20 edges of these two polygons together with an additional 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Petersen Graph
In graph theory, the generalized Petersen graphs are a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. They include the Petersen graph and generalize one of the ways of constructing the Petersen graph. The generalized Petersen graph family was introduced in 1950 by H. S. M. Coxeter and was given its name in 1969 by Mark Watkins. Definition and notation In Watkins' notation, ''G''(''n'', ''k'') is a graph with vertex set :\ and edge set :\ where subscripts are to be read modulo ''n'' and ''k'' < ''n''/2. Some authors use the notation ''GPG''(''n'', ''k''). Coxeter's notation for the same graph would be + , a combination of the Schläfli symbols for the regular ''n''-gon and star polygon from which the graph is formed. The Petersen graph itself is ''G''(5, 2) or + . Any generalized Petersen graph can also be constructed from a voltage graph with two vertices, two self-loops, and one other ed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Isomorphism
In graph theory, an isomorphism of graphs ''G'' and ''H'' is a bijection between the vertex sets of ''G'' and ''H'' : f \colon V(G) \to V(H) such that any two vertices ''u'' and ''v'' of ''G'' are adjacent in ''G'' if and only if f(u) and f(v) are adjacent in ''H''. This kind of bijection is commonly described as "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection. If an isomorphism exists between two graphs, then the graphs are called isomorphic and denoted as G\simeq H. In the case when the bijection is a mapping of a graph onto itself, i.e., when ''G'' and ''H'' are one and the same graph, the bijection is called an automorphism of ''G''. If a graph is finite, we can prove it to be bijective by showing it is one-one/onto; no need to show both. Graph isomorphism is an equivalence relation on graphs and as such it partitions the class of all graphs into equivalence classes. A set of graphs isomorphic to each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Graph
In the mathematical field of graph theory, a graph is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices and of , there is an automorphism :f : V(G) \rightarrow V(G) such that :f(u_1) = u_2 and f(v_1) = v_2. In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called -transitive or flag-transitive. By definition (ignoring and ), a symmetric graph without isolated vertices must also be vertex-transitive. Since the definition above maps one edge to another, a symmetric graph must also be edge-transitive. However, an edge-transitive graph need not be symmetric, since might map to , but not to . Star graphs are a simple example of being edge-transitive without being vertex-transitive or symmetric. As a further example, semi-symmetric graphs are edge-transitive and regular, but not vertex-transitiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathem ... [...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 H. S. M. Coxeter and Robert Frucht, for the representation of cubic graphs that contain a Hamiltonian cycle. The cycle itself includes two out of the three adjacencies for each 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 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 numbers between the brackets are interpreted modulo ''N'', where ''N'' is the number of vertic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Graph
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. Determining whether such paths and cycles exist in graphs (the Hamiltonian path problem and Hamiltonian cycle problem) are NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as ''Hamilton's puzzle'', which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Subgraph
In the mathematical field of graph theory, an induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and ''all'' of the edges (from the original graph) connecting pairs of vertices in that subset. Definition Formally, let G=(V,E) be any graph, and let S\subset V be any subset of vertices of . Then the induced subgraph G is the graph whose vertex set is S and whose edge set consists of all of the edges in E that have both endpoints in S . That is, for any two vertices u,v\in S , u and v are adjacent in G if and only if they are adjacent in G . The same definition works for undirected graphs, directed graphs, and even multigraphs. The induced subgraph G may also be called the subgraph induced in G by S , or (if context makes the choice of G unambiguous) the induced subgraph of S . Examples Important types of induced subgraphs include the following. *Induced paths are induced subgraphs that are paths. The shortest path between ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kneser Graph
In graph theory, the Kneser graph (alternatively ) is the graph whose vertices correspond to the -element subsets of a set of elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint. Kneser graphs are named after Martin Kneser, who first investigated them in 1956. Examples The Kneser graph is the complete graph on vertices. The Kneser graph is the complement of the line graph of the complete graph on vertices. The Kneser graph is the odd graph ; in particular is the Petersen graph (see top right figure). The Kneser graph , visualized on the right. Properties Basic properties The Kneser graph K(n,k) has \tbinom vertices. Each vertex has exactly \tbinom neighbors. The Kneser graph is vertex transitive and arc transitive. When k=2, the Kneser graph is a strongly regular graph, with parameters ( \tbinom, \tbinom, \tbinom, \tbinom ). However, it is not strongly regular when k>2, as different pairs of nonadjacent verti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Double Cover
In graph theory, the bipartite double cover of an undirected graph is a bipartite, covering graph of , with twice as many vertices as . It can be constructed as the tensor product of graphs, . It is also called the Kronecker double cover, canonical double cover or simply the bipartite double of . It should not be confused with a cycle double cover of a graph, a family of cycles that includes each edge twice. Construction The bipartite double cover of has two vertices and for each vertex of . Two vertices and are connected by an edge in the double cover if and only if and are connected by an edge in . For instance, below is an illustration of a bipartite double cover of a non-bipartite graph . In the illustration, each vertex in the tensor product is shown using a color from the first term of the product () and a shape from the second term of the product (); therefore, the vertices in the double cover are shown as circles while the vertices are shown as squares. : T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Desargues' Theorem
In projective geometry, Desargues's theorem, named after Girard Desargues, states: :Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''. Denote the three vertices of one triangle by and , and those of the other by and . ''Axial perspectivity'' means that lines and meet in a point, lines and meet in a second point, and lines and meet in a third point, and that these three points all lie on a common line called the ''axis of perspectivity''. ''Central perspectivity'' means that the three lines and are concurrent, at a point called the ''center of perspectivity''. This intersection theorem is true in the usual Euclidean plane but special care needs to be taken in exceptional cases, as when a pair of sides are parallel, so that their "point of intersection" recedes to infinity. Commonly, to remove these exceptions, mathematicians "complete" the Euclidean plane by adding points at infinity, following Jean-Victor Poncelet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perspective (geometry)
Two figures in a plane are perspective from a point ''O'', called the center of perspectivity if the lines joining corresponding points of the figures all meet at ''O''. Dually, the figures are said to be perspective from a line if the points of intersection of corresponding lines all lie on one line. The proper setting for this concept is in projective geometry where there will be no special cases due to parallel lines since all lines meet. Although stated here for figures in a plane, the concept is easily extended to higher dimensions. Terminology The line which goes through the points where the figure's corresponding sides intersect is known as the axis of perspectivity, perspective axis, homology axis, or archaically, perspectrix. The figures are said to be perspective from this axis. The point at which the lines joining the corresponding vertices of the perspective figures intersect is called the center of perspectivity, perspective center, homology center, pole, or archaica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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