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Dejter Graph
In the mathematical field of graph theory, the Dejter graph is a 6-regular graph with 112 vertices and 336 edges.Borges J.; Dejter I. J. "On perfect dominating sets in hypercubes and their complements", J. Combin. Math. Combin. Comput. 20 (1996), 161-173Dejter I. J. "Symmetry of factors of the 7-cube Hamming shell", J. Combin. Des. 5 (1997), 301–309Dejter I. J.; Guan P. "Square-blocking edge subsets in hypercubes and vertex avoidance", Graph theory, combinatorics, algorithms, and applications (San Francisco, CA, 1989), 162–174, SIAM, Philadelphia, PA, 1991Dejter I. J.; Pujol J. "Perfect domination and symmetry in hypercubes", Proceedings of the Twenty-Sixth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, Florida, 1995). Congr. Numer. 111 (1995), 18–32Dejter I. J.; Weichsel P. M. "Twisted perfect dominating subgraphs of hypercubes", Proceedings of the Twenty-Fourth Southeastern International Conference on Combinatorics, Graph ... [...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. 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-free configuration with 56 lines and 56 points. In this configuration, each line contains exactly 3 points, each point belongs to exactly 3 lines and any two lines intersect in at most one point. Algebraic properties The automorphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets U and V, that is every edge connects a vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets U and V may be thought of as a coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a triangle: after one node is colored blue and another red, the third vertex of the triangle is connected to vertices of both colors, preventing it from being assigned either color. One often writes G=(U,V,E) to denote a bipartite graph whose partition has the parts U and V, with E denotin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heawood Graph
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Heawood is a surname. Notable people with the surname include: * Jonathan Heawood, British journalist * Percy John Heawood (1861–1955), British mathematician ** Heawood conjecture ** Heawood graph ** Heawood number See also *Heywood (surname) Heywood is a surname. Notable people with the surname include: * Abel Heywood (1810–1893), English politician * Angela Heywood (1840–1935), United States suffragist, socialist, abolitionist * Anne Heywood (born 1932), British film actress * Art ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covering Graph
In the mathematical discipline of graph theory, a graph is a covering graph of another graph if there is a covering map from the vertex set of to the vertex set of . A covering map is a surjection and a local isomorphism: the neighbourhood of a vertex in is mapped bijectively onto the neighbourhood of in . The term lift is often used as a synonym for a covering graph of a connected graph. Though it may be misleading, there is no (obvious) relationship between covering graph and vertex cover or edge cover. The combinatorial formulation of covering graphs is immediately generalized to the case of multigraphs. In the case of a multigraph with a 1-dimensional cell complex, a covering graph is nothing but a special example of covering spaces of topological spaces, so the terminology in the theory of covering spaces is available; say covering transformation group, universal covering, abelian covering, and maximal abelian covering. Definition Let ''G'' = (''V''1, ''E''1) ... [...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- ... [...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] |
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- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Factorization
In graph theory, a factor of a graph ''G'' is a spanning subgraph, i.e., a subgraph that has the same vertex set as ''G''. A ''k''-factor of a graph is a spanning ''k''-regular subgraph, and a ''k''-factorization partitions the edges of the graph into disjoint ''k''-factors. A graph ''G'' is said to be ''k''-factorable if it admits a ''k''-factorization. In particular, a 1-factor is a perfect matching, and a 1-factorization of a ''k''-regular graph is an edge coloring with ''k'' colors. A 2-factor is a collection of cycles that spans all vertices of the graph. 1-factorization If a graph is 1-factorable (ie, has a 1-factorization), then it has to be a regular graph. However, not all regular graphs are 1-factorable. A ''k''-regular graph is 1-factorable if it has chromatic index ''k''; examples of such graphs include: * Any regular bipartite graph. Hall's marriage theorem can be used to show that a ''k''-regular bipartite graph contains a perfect matching. One can then remove the ... [...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-transiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercube Graph
In graph theory, the hypercube graph is the graph formed from the vertices and edges of an -dimensional hypercube. For instance, the cube graph is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. has vertices, edges, and is a regular graph with edges touching each vertex. The hypercube graph may also be constructed by creating a vertex for each subset of an -element set, with two vertices adjacent when their subsets differ in a single element, or by creating a vertex for each -digit binary number, with two vertices adjacent when their binary representations differ in a single digit. It is the -fold Cartesian product of the two-vertex complete graph, and may be decomposed into two copies of connected to each other by a perfect matching. Hypercube graphs should not be confused with cubic graphs, which are graphs that have exactly three edges touching each vertex. The only hypercube graph that is a cubic graph is the cubical graph . Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
