selfref For information about graphs in Wikipedia, see Wikipedia Graphs . wiktionary Graph may refer to A Information graphics graphic such as a chart or diagram depicting the relationship between two or more variables used, for instance, in visualising scientific data. In mathematics Graph mathematics , is a set of vertices and edges. Graph theory Graph of a function In computer science Graph data structure , an abstract data type representing relationships or connections Graph software , the name of a software application for mathematical plotting Conceptual graph , a model for knowledge representation and reasoning Other uses HMS Graph P715 , a submarine of the Royal Navy United Kingdom See also Grapheme linguistics wiktionary graphy graphy suffix Latin for to write or draw Graf Graff disambiguation List of information graphics software disambig DEFAULTSORT de Graph es Grafo desambiguaci n eu Grafo argipena fr Graphe ms Graf ja ru uk ur Graph ... more details
Orphan date November 2006 Image s graph.gif right thumb 275px Visual representation of an S graph to efficiently solving batch process scheduling problems in chemical plant s. ref Cite journal unused data Hungarian Journal for Industrial Chemistry last Holczinger first T. coauthors J Romero, L Puigjaner, F Friedler title Scheduling of Multipurpose Batch Processes with Multiple Batches of the Products volume 30 pages 305 312 date 2002 12 02 ref ref name AICE Cite journal last Romero first Javier coauthors Luis Puigjaner, Tibor Holczinger, Ferenc Friedler title Scheduling intermediate storage multipurpose batch plants using the S graph journal American Institute of Chemical Engineers volume 50 issue 2 pages 403 417 date 2004 02 18 ref S graph is especially developed for the problems with non intermediate storage NIS policy, which often appears in chemical productions, but it is also capable to solve problems with unlimited intermediate storage UIS policy. ref name AICE Overview S graph representation has the advantage of exploiting problem specific knowledge to develop efficient scheduling algorithm s. ref name AICE There are products, and a set of task, which have to be performed to produce a product. There are dependencies between the tasks, and every task has a set of equipments, that can perform the task. Different processing times can be set for the same task in different equipments. It is also possible to have more equipment units from the same type, or define changeover times between two task in one equipment. There are two types of the scheduling problems The number of batches to produce is set, and we try to minimize the makespan processing time . Every product has a revenue, and a time horizon is set. The objective is to maximize the revenue in this fixed time horizon. S graph framework also contains Combinatorics combinatoric algorithm s to solve both of these problems. References Reflist External links http www.s graph.com S graph website Category Job scheduling ... more details
In mathematics , a convex graph may be a convex bipartite graph a convex plane graph the graph of a function graph of a convex function disambig ... more details
Primal graph may refer to Primal graph hypergraphs of a hypergraph A primal graph may be the planar graph from which a dual graph is formed Primal constraint graph disambig ... more details
wiktionarypar Periodic Graph Periodic graph periodic graph Periodic graphs periodic graphs Periodic graph can mean Periodic graph crystallography or crystal net , a Euclidean graph representing the atomic or molecular structure of a crystal. Periodic graph geometry , a Euclidean graph preserved under a lattice of translations. Periodic graphgraph theory , a graph that is periodic with respect to a graph theoretic operator disambig ... more details
Other uses Periodic graph disambiguation Periodic graph In graph theory , a branch of mathematics, a periodic graph with respect to an operator F on graphs is one for which there exists an integer n 0 such that F sup n sup G is graph isomorphism isomorphic to G . ref Citation last Zelinka first B. title Periodicity of graph operators journal Discrete Mathematics volume 235 pages 349 351 year 2001 url http www.sciencedirect.com science? ob ArticleURL& udi B6V00 433PBV1 16& user 10& coverDate 05 2F28 2F2001& rdoc 34& fmt high& orig browse& srch doc info 23toc 235632 232001 23997649998 23251347 23FLT 23display 23Volume & cdi 5632& sort d& docanchor & ct 39& acct C000050221& version 1& urlVersion 0& userid 10&md5 c91abbf2a679877d22212fa49932088c accessdate 14 August 2010 ref For example, every graph is periodic with respect to the complement graph complementation operator , whereas only complete graph s are periodic with respect to the operator that assigns to each graph the complete graph on the same vertices. Periodicity is one of many properties of graph operators, the central topic in graph dynamics . ref Cite book last Prisner first Erich title Graph Dynamics publisher CRC Press year 1995 isbn 9780582286962 ref References Reflist DEFAULTSORT Periodic GraphGraph Theory Category Graph invariants Category Graph operations math stub ... more details
Citations missing date October 2008 In the mathematics mathematical field of graph theory , a quartic graph is a graph mathematics graph where all vertex graph theory vertices have degree graph theory degree 4. In other words a quartic graph is a 4 regular graph . A biquartic graph is a quartic bipartite graph . It is an open conjecture that all quartic graphs have an even number of Hamiltonian circuit s. It is known that quartic graphs have an even number of Hamiltonian decomposition s. See also Cubic graph DEFAULTSORT Quartic Graph Category Graph families Category Regular graphs ... more details
In mathematics , and, in particular, in graph theory , a rooted graph is a graph mathematics mathematical graph in which one node graph theory node is labelled in a special way to distinguish it from the graph s other nodes. This special node is called the root of the graph. The number of rooted graphs for 1, 2, ... nodes is 1, 2, 6, 20, 90, 544, ... OEIS id A000666 A special case of interest are rooted tree s. External links http mathworld.wolfram.com RootedGraph.html MathWorld Rooted graph Combin stub Category Extensions and generalizations of graphs ... more details
infobox graph name Shrikhande graph image Image Shrikhande graph square.svg 250px image caption The Shrikhande graph namesake S. S. Shrikhande vertices 16 edges 48 chromatic number 4 chromatic index 6 automorphisms 192 diameter 2 radius 2 girth 3 properties Strongly regular graph Strongly regular br Hamiltonian graph Hamiltonian br Symmetric graph Symmetric br Eulerian graph Eulerian br Integral graph Integral In the mathematics mathematical field of graph theory , the Shrikhande graph is a Gallery of named graphs named graph discovered by S. S. Shrikhande in 1959. ref mathworld urlname ShrikhandeGraph title Shrikhande Graph ref ref citation first S. S. last Shrikhande authorlink S. S. Shrikhande ... regular graph with 16 vertex graph theory vertices and 48 edge graph theory edges , with each vertex having a degree graph theory degree of 6. Properties In the Shrikhande graph, any two vertices I and J ... are 16,6,2,2 , with math lambda mu 2 math , this equality implying that the graph is associated with a symmetry symmetric BIBD . It shares these parameters with a different graph, the 4× 4 rook s graph . The Shrikhande graph is Neighbourhood graph theory locally hexagonal that is, the neighbors of each vertex form a cycle graph cycle of six vertices. As with any locally cyclic graph, the Shrikhande graph is the n skeleton 1 skeleton of a Triangulation topology Whitney triangulation of some surface in the case of the Shrikhande graph, this surface is a torus in which each vertex is surrounded ... Shrikhande graph . ref Thus, the Shrikhande graph is a toroidal graph . The dual of this embedding is the Dyck graph , a cubic symmetric graph. The Shrikhande graph is not a distance transitive graph . It is the smallest distance regular graph that is not distance transitive. ref citation last1 ... location New York publisher Springer Verlag pages 104 105 and 136 year 1989 . ref The Graph automorphism automorphism group of the Shrikhande graph is of order 192. It acts transitively on the vertices ... more details
In the mathematics mathematical field of graph theory , an integral graph is a graph whose Spectral graph theory spectrum consists entirely of integers. In other words, a graphs is an integral graph if all the eigenvalues of its characteristic polynomial are integers. ref MathWorld urlname IntegralGraph title Integral Graph ref The notion was introduced in 1974 by Harary and Schwenk. ref Harary, F. and Schwenk, A. J. Which Graphs have Integral Spectra? In Graphs and Combinatorics Ed. R. Bari and F. Harary . Berlin Springer Verlag, pp. 45&ndash 51, 1974. ref Examples The complete graph K sub n sub is integral for all n . The edgeless graph math bar K n math is integral for all n . Among the cubics symmetric graphs the utility graph , the Petersen graph , the Nauru graph and the Desargues graph are integral. The Higman Sims graph , the Hall Janko graph , the Clebsch graph , the Hoffman Singleton graph , the Shrikhande graph and the Hoffman graph are integral. References reflist Category Graph families Category Algebraic graph theory es Grafo integral fr Graphe int gral pt Grafo integral ... more details
The terms lattice graph , mesh graph , or grid graph refer to a number of categories of graph mathematics graph s graph drawing whose drawing corresponds to some grid mesh lattice, i.e., its vertices correspond to the nodes of the mesh and its edges correspond to the ties between the nodes. Square grid graph A common type of a lattice graph known under different names, such as square grid graph is the graph whose vertices correspond to the points in the plane with integer coordinates, x coordinates being in the range 0,..., n, y coordinates being in the range 1,...m, and two vertices are connected by an edge whenever the corresponding points are at distance 1. In other words, it is a unit distance graph for the described point set. ref name weiss Properties A square grid graph is a Cartesian product of graphs , namely, of two path graph s with n and m edges. ref name weiss Since a path graph is a median graph , the latter fact implies that the square grid graph is also a median graph. All grid graphs are bipartite graph bipartite . A path graph may also be considered to be a grid graph on the grid n times 1. A 2x2 grid graph is a cycle graph 4 cycle . ref name weiss CRC Concise Encyclopedia of Mathematics , by Eric W. Weisstein , article Grid graph mathworld urlname GridGraph title Grid graph ref Other kinds A triangular grid graph is a graph that corresponds to a triangular grid. A Hanan grid graph for a finite set of points in the plane is produced by the grid obtained by intersections of all vertical and horizontal lines through each point of the set. The rook s graph the graph that represents all legal moves of the Rook chess rook chess Chess piece piece on a chessboard is also sometimes called the lattice graph. References reflist Category Planar graphs Category Graph families ... more details
infobox graph name Dipole graph image Image Dipole graph.svg 140px image caption vertices 2 edges n chromatic number 2 chromatic index n diameter 1 In graph theory , a dipole graph or dipole is a multigraph consisting of two vertex graph theory vertices connected with a number of edge graph theory edges . A dipole graph containing n edges is called the order n dipole graph, and is denoted by D sub n sub . The order n dipole graph is dual graph dual to the cycle graph C sub n sub . References MathWorld title Dipole Graph urlname DipoleGraph Jonathan L. Gross and Jay Yellen, 2006. Graph Theory and Its Applications, 2nd Ed. , p. 17. Chapman & Hall CRC. ISBN 1 58488 505 X Combin stub Category Extensions and generalizations of graphs Category Parametric families of graphs Category Regular graphs ... more details
Infobox graph name Butterfly graph image Image Butterfly graph.svg 200px vertices 5 edges 6 automorphisms ... planar graph Planar br unit distance graph Unit distance br Eulerian graph Eulerian In the mathematics mathematical field of graph theory , the butterfly graph also called the bowtie graph and the hourglass graph is a planar graph planar undirected graph with 5 vertices and 6 edges. ref MathWorld urlname ButterflyGraph title Butterfly Graph ref ref ISGCI Information System on Graph Class ... . ref It can be constructed by joining 2 copies of the cycle graph C sub 3 sub with a common vertex and is therefore isomorphic to the friendship graph F sub 2 sub . The butterfly Graph has graph diameter diameter 2 and girth graph theory girth 3, radius 1, chromatic number 3, chromatic index 4 and is both Eulerian graph Eulerian and unit distance graph unit distance . It is also a 1 k vertex connected graph vertex connected graph and a 2 k edge connected graph edge connected graph . There are only 3 Graceful labeling non graceful simple graphs with five vertices. One of them is the butterfly graph. The two others are cycle graph C sub 5 sub and the complete graph K sub 5 sub . ref name Mat2007 mathworld title Graceful graph urlname GracefulGraph ref Bowtie free graphs A graph is bowtie free if it has no butterfly as an induced subgraph . The triangle free graph s are bowtie free graphs, since every butterfly contains a triangle. In a k vertex connected graph k vertex connected graph, and edge is said k contractible if the contraction of the edge results in a k connected graph. Ando, Kaneko, Kawarabayashi and Yoshimoto proved that every k vertex connected bowtie free graph has a k contractible edge. ref Kiyoshi Ando Contractible Edges in a k Connected Graph ... The full automorphism group of the butterfly graph is a group of order 8 isomorphic to the Dihedral ... and reflections. The characteristic polynomial of the butterfly graph is math x 1 x 1 2 x 2 x 4 ... more details
infobox graph name Harries graph image Image Harries graph.svg 220px image caption The Harries graph namesake vertices 70 edges 105 automorphisms 120 Symmetric group S sub 5 sub girth 10 diameter 6 radius 6 chromatic number 2 chromatic index 3 properties Cubic graph Cubic br Cage graph theory Cage br Triangle free graph Triangle free br Hamiltonian graph Hamiltonian In the mathematics mathematical field of graph theory , the Harries graph or Harries 3 10 cage is a 3 regular graph regular undirected graph with 70 vertices and 105 edges. ref MathWorld urlname HarriesGraph title Harries Graph ref The Harries graph has chromatic number 2, chromatic index 3, radius 6, diameter 6, girth 10 and is Hamiltonian graph Hamiltonian . It is also a 3 k vertex connected graph vertex connected and 3 k edge connected graph edge connected planar graph non planar cubic graph . The characteristic polynomial of the Harries graph is math x 3 x 1 4 x 1 4 x 3 x 2 6 x 2 2 x 4 6x 2 2 5 x 4 6x 2 3 4 x 4 6x 2 6 5. , math History In 1972, A. T. Balaban published a 3 10 cage graph , a cubic graph that has as few vertices as possible for girth 10. ref A. T. Balaban, A trivalent graph of girth ten, J. Combin. Theory Ser. B 12, 1 5. 1972. ref It was the first 3 10 cage discovered but it was not unique. ref Pisanski ... was given by O Keefe and Wong in 1980. ref M. O Keefe and P.K. Wong, A smallest graph of girth 10 ... the Balaban 10 cage , the Harries graph and the Harries Wong graph . ref Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York North Holland, p. 237, 1976. ref Moreover, the Harries Wong graphgraph and Harries graph are Spectral graph theory cospectral graphs . Gallery gallery Image Harries graph 2COL.svg The chromatic number of the Harries graph is 2. Image Harries graph 3color edge.svg The chromatic index of the Harries graph is 3. Image harries graph alternative drawing.svg Alternative drawing of the Harries graph. gallery References reflist Category Individual ... more details
infobox graph name Pappus graph image Image Pappus graph LS.svg 250px image caption The Pappus graph ... number 2 chromatic index 3 properties Symmetric graph Symmetric br Distance transitive graph Distance transitive br Distance regular graph Distance regular br Cubic graph Cubic br Hamiltonian graph Hamiltonian In the mathematics mathematical field of graph theory , the Pappus graph is a 3 regular graph regular undirected graph with 18 vertices and 27 edges, formed as the Levi graph of the Pappus configuration ref MathWorld urlname PappusGraph title Pappus Graph ref . It is named after Pappus ... the hexagon theorem describing the Pappus configuration. All the cubic graph cubic distance regular graph s are known the Pappus graph is one of the 13 such graphs. ref Brouwer, A. E. Cohen, A. M. and Neumaier, A. Distance Regular Graphs. New York Springer Verlag, 1989. ref The Pappus graph has Crossing number graph theory rectilinear crossing number 5, and is the smallest cubic graph with that crossing number OEIS id A110507 . It has Girth graph theory girth 6, diameter 4, radius 4, chromatic number 2, chromatic index 3 and is both 3 k vertex connected graph vertex connected and 3 k edge connected graph edge connected . The Pappus graph has a chromatic polynomial equal to math x 1 x x ... graph has also been used to refer to a related nine vertex graph ref citation last Kagno first I ... pair of points on the same line this nine vertex graph is 6 regular, and is the complement graph of the union of three disjoint triangle graph s. Algebraic properties The automorphism group of the Pappus graph is a group of order 216. It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Pappus graph is a symmetric graph . It has automorphisms that take any ... graph, referenced as F018A, is the only cubic symmetric graph on 18 vertices. ref Royle, G. http .... Comput. 40, 41 63, 2002. ref The characteristic polynomial of the Pappus graph is math x 3 x 4 x 3 ... more details
infobox graph name Franklin Graph image Image Franklin graph hamiltonian.svg 220px image caption The Franklin Graph namesake Philip Franklin vertices 12 edges 18 automorphisms 48 Cyclic group Z 2 Z Symmetric group S sub 4 sub girth 4 radius 3 diameter 3 chromatic number 2 chromatic index 3 properties Cubic graph Cubic br Hamiltonian graph Hamiltonian br Bipartite graph Bipartite br Triangle free graph Triangle free br Perfect graph Perfect br vertex transitive graph Vertex transitive In the mathematics mathematical field of graph theory , the Franklin graph a 3 regular graph with 12 vertices and 18 edges. ref MathWorld urlname FranklinGraph title Franklin Graph ref The Franklin graph is named after Philip Franklin , who disproved the Heawood conjecture on the number of colors needed when a two dimensional surface is partitioned into cells by a graph embedding . ref MathWorld urlname HeawoodConjecture ... graph can be embedded onto the Klein bottle so that it forms a map requiring six colors, showing that six colors are sometimes necessary in this case. It is Hamiltonian graph Hamiltonian and has chromatic number 2, chromatic index 3, radius 3, diameter 3 and girth graph theory girth 4. It is also a 3 k vertex connected graph vertex connected and 3 k edge connected graph edge connected perfect graph . Algebraic properties The Graph automorphism automorphism group of the Franklin graph is of order ... group Z 2 Z and the symmetric group S sub 4 sub . It acts transitively on the vertices of the graph, making it vertex transitive graph vertex transitive . The characteristic polynomial of the Franklin graph is math x 3 x 1 3 x 1 3 x 3 x 2 3 2. math Gallery gallery Image Franklin graph 2COL.svg The chromatic number of the Franklin graph is 2. Image Franklin graph 3color edge.svg The chromatic index of the Franklin graph is 3. Image franklin graph.svg Alternative drawing of the Franklin graph gallery References reflist DEFAULTSORT Franklin Graph Category Individual graphs Category ... more details
infobox graph name Hoffman graph image Image Hoffman graph.svg 220px image caption The Hoffman graph namesake Alan Hoffman mathematician Alan Hoffman vertices 16 edges 32 automorphisms 48 Z 2 Z S sub 4 sub girth 4 diameter 4 radius 4 chromatic number 2 chromatic index 4 properties Hamiltonian graph Hamiltonian ref MathWorld urlname HamiltonianGraph title Hamiltonian Graph ref br Bipartite graph Bipartite br Perfect graph Perfect br Eulerian graph Eulerian In the mathematics mathematical field of graph theory , the Hoffman graph is a 4 regular graph with 16 vertices and 48 edges discovered by Alan Hoffman mathematician Alan Hoffman . ref MathWorld urlname HoffmanGraph title Hoffman graph ref Published in 1963, it is cospectral to the hypercube graph Q sub 4 sub . ref Hoffman, A. J. On the Polynomial of a Graph. Amer. Math. Monthly 70, 30 36, 1963. ref ref van Dam, E. R. and Haemers, W. H. Spectral Characterizations of Some Distance Regular Graphs. J. Algebraic Combin. 15, 189 202, 2003. ref The Hoffman graph has many common properties with the hypercube Q sub 4 sub both are Hamiltonian graph Hamiltonian and have chromatic number 2, chromatic index 4, radius 4, girth 4 and diameter 4. It is also a 4 k vertex connected graph vertex connected graph and a 4 k edge connected graph edge connected graph . Algebraic properties The Hoffman graph is not a vertex transitive graph and its full automorphism group is a group of order 48 isomorphic to the direct product of groups direct product ... graph is equal to math x 4 x 2 4 x 6 x 2 4 x 4 math making it an integral graph a graph whose Spectral graph theory spectrum consists entirely of integers. It is the same spectrum than the hypercube Q sub 4 sub . Gallery gallery Image Hoffman graph hamiltonian.svg The Hoffman graph is Hamiltonian graph Hamiltonian . Image Hoffman graph 2COL.svg The chromatic number of the Hoffman graph is 2. Image Hoffman graph 4color edge.svg The chromatic index of the Hoffman graph is 4. gallery ... more details
infobox graph name King s graph image Image King s graph.svg 180px image caption 8x8 King s graph vertices nm edges 4 nm 3 n m 2 chromatic number chromatic index girth properties In graph theory , a king s graph is a Graph mathematics graph that represents all legal moves of the king chess king chess chess piece piece on a chessboard where each vertex represents a square on a chessboard and each edge is a legal move. More specifically, an math n times m math king s graph is a king s graph of an math n times m math chessboard. For a math n times m math king s graph the total number of vertices is simply math n m math . For a math n times n math king s graph the total number of vertices is simply math n 2 math and the total number of edges is math 2n 2 2n 1 math . Additionally, the number of edges for various math n math is identified as OEIS2C id A002943 in the On Line Encyclopedia of Integer Sequences . Neighbourhood graph theory Neighbourhood in the king s graph corresponds to the Moore neighborhood for cellular automata. See also Knight s graph Rook s graph Lattice graph Category Chess and mathematics Category Parametric families of graphs ... more details
infobox graph name Null graph vertices 0 edges 0 automorphisms 1 In the mathematics mathematical field of graph theory , the null graph or the empty graph is either the graph mathematics graph with no vertices and hence no edges, or any graph with no edges. The null graph in the former sense is the initial object in the category mathematics category of graphs, according to some definitions of a category of graphs. Having no vertices, the null graph therefore also has no Connected component graph theory connected components . Thus, although the null graph is a Tree graph theory forest a graph with no cycles , it is not a Tree graph theory tree , as trees have one connected component. Edgeless graph infobox graph name Edgeless graph vertices n edges 0 automorphisms n chromatic number 1 notation math bar K n math properties Integral graph Integral br Symmetric graph Symmetric Some authors feel that a better term for the latter sense V , for any set V is the more explicit edgeless graph . This reserves the term null graph for the former sense a graph without even any vertices. Still others make this distinction by applying the label empty to these graphs with no edges. ref MathWorld urlname EmptyGraph title Empty Graph ref ref MathWorld urlname NullGraph title Null Graph ref The n vertex edgeless graph is the complement graph for the complete graph math K n math , and therefore it is commonly denoted as math bar K n math . Even though this definition provides a solid basis for defining certain operations on graphs e.g. decomposition considering graphs as sets of vertices and edges V , E this definition raises a problem in uniqueness of the null element of graphs. See also Glossary of graph theory Cycle graph Path graph Notes reflist References commonscat Null graphs Frank Harary Harary, F. and Read, R. 1973 , Is the null graph a pointless concept? , Graphs and Combinatorics Conference, George Washington University , Springer Verlag, New York, NY. Category Individual graphs ... more details
infobox graph name Robertson graph image Image Robertson graph hamiltonian.svg 240px image caption The Robertson graph is Hamiltonian. namesake Neil Robertson mathematician Neil Robertson vertices 19 edges 38 automorphisms 24 dihedral group D sub 12 sub girth 5 diameter 3 radius 3 chromatic number 3 chromatic index 5 ref MathWorld urlname Class2Graph title Class 2 Graph ref properties Cage graph theory Cage br Hamiltonian graph Hamiltonian In the mathematics mathematical field of graph theory , the Robertson graph or 4,5 cage is a 4 regular graph regular undirected graph with 19 vertices and 38 edges named after Neil Robertson mathematician Neil Robertson . ref MathWorld urlname RobertsonGraph title Robertson Graph ref ref Bondy, J. A. and Murty, U. S. R. Graph Theory with Applications. New York North Holland, p. 237, 1976. ref The Robertson graph is the unique cage graph 4,5 cage graph and was discovered by Robertson in 1964. ref Robertson, N. The Smallest Graph of Girth 5 and Valency 4. Bull. Amer. Math. Soc. 70, 824 825, 1964. ref As a cage graph, it is the smallest 4 regular graph with girth 5. It has chromatic number 3, chromatic index 5, diameter 3, radius 3 and is both 4 k vertex connected graph vertex connected and 4 k edge connected graph edge connected . The Robertson graph is also a Hamiltonian graph which possesses formatnum 5376 distinct directed Hamiltonian cycles. Algebraic properties The Robertson graph is not a vertex transitive graph and its full automorphism group is isomorphic to the dihedral group of order 24, the group of symmetries of an regular dodecagon ... cage survey, Electr. J. Combin. 15, 2008. ref The characteristic polynomial of the Robertson graph ... Robertson graph.svg The Robertson graph as drawn in the original publication. Image Robertson graph 3COL.svg The chromatic number of the Robertson graph is 3. Image Robertson graph 5color edge.svg The chromatic index of the Robertson graph is 5. gallery References reflist Category Individual ... more details
infobox graph name Folkman graph image Image Folkman graph alt.svg 220px image caption The Folkman graph namesake J. Folkman vertices 20 edges 40 girth 4 diameter 4 radius 3 chromatic number 2 chromatic index 4 properties Hamiltonian graph Hamiltonian br regular graph Regular br Bipartite graph Bipartite br semi symmetric graph Semi symmetric br Eulerian graph Eulerian br perfect graph Perfect In the mathematics mathematical field of graph theory , the Folkman graph , named after Jon Folkman , is a Bipartite graph bipartite 4 regular graph regular graph with 20 vertex graph theory vertices and 40 edges. ref MathWorld title Folkman graph urlname FolkmanGraph ref The Folkman graph is Hamiltonian graph Hamiltonian and has chromatic number 2, chromatic index 4, radius 3, diameter 4 and girth graph theory girth 4. It is also a 4 k vertex connected graph vertex connected and 4 k edge connected graph edge connected perfect graph . Algebraic properties The automorphism group of the Folkman graphgraph acts transitively on its edges but not on its vertices. It is the smallest undirected graph that is edge transitive graph edge transitive and regular, but not vertex transitive graph vertex transitive . ref Skiena, S. Implementing Discrete Mathematics Combinatorics and Graph Theory with Mathematica. Reading, MA Addison Wesley, pp. 186 187, 1990 ref Such graphs are called semi symmetric graph s and were first studied by Folkman in 1967 who discovered the graph on 20 vertices that now ... 10.1016 S0021 9800 67 80069 3 ref As a semi symmetric graph, the Folkman graph is bipartite graph bipartite .... In the diagram below indicating the chromatic number of the graph, the green vertices can not be mapped ... graph is math x 4 x 10 x 4 x 2 6 4 math . Gallery gallery Image Folkman graph 4color edge.svg The chromatic index of the Folkman graph is 4. Image Folkman graph.svg The chromatic number of the Folkman graph is 2. File Folkman Lombardi.svg The Folkman graph is Hamiltonian graph Hamiltonian ... more details
infobox graph name Holt graph image File Holt graph.svg 220px image caption In the Holt graph, all vertices ... 3 chromatic number 3 chromatic index 5 properties Vertex transitive graph Vertex transitive br Edge transitive graph Edge transitive br Half transitive graph Half transitive br Hamiltonian graph Hamiltonian br Eulerian graph Eulerian br Cayley graph In the mathematics mathematical field of graph theory , the Holt graph or Doyle graph is the smallest half transitive graph , that is, the smallest example of a Vertex transitive graph vertex transitive and Edge transitive graph edge transitive graph which is not also symmetric graph symmetric . ref Doyle, P. A 27 Vertex Graph That Is Vertex Transitive ..., Handbook of Graph Theory , CRC Press, 2004, ISBN 1584880902, p. 491. ref It is named after Peter G. Doyle and Derek F. Holt, who discovered the same graph independently in 1976 ref citation first P ... . As cited by MathWorld. ref and 1981 ref name holt citation title A graph which is edge transitive but not arc transitive first Derek F. last Holt journal Journal of Graph Theory volume 5 issue 2 pages 201 204 year 1981 doi 10.1002 jgt.3190050210 . ref respectively. The Holt Graph has graph diameter diameter 3, radius 3 and girth graph theory girth 5, chromatic number 3, chromatic index 5 and is Hamiltonian graph Hamiltonian with formatnum 98472 distinct Hamiltonian cycles. ref name Mathworld MathWorld urlname DoyleGraph title Doyle Graph ref It is also a 4 k vertex connected graph vertex connected and a 4 k edge connected graph edge connected graph. It has an graph ... group than a symmetric graph with the same number of vertices and edges would have. The graph drawing ... of the Holt graph is math x 3 6x 2 6 x 2 4 x 1 4 x 4 . math Gallery gallery Image Holt graph 3COL.svg The chromatic number of the Holt graph is 3. Image Holt graph 5color edge.svg The chromatic index of the Holt graph is 5. Image Holt graph hamiltonian.svg The Holt graph is Hamiltonian ... more details
Infobox graph name Clebsch graph image Image Clebsch graph.svg 200px image caption A drawing of the Clebsch graph namesake Alfred Clebsch vertices 16 edges 40 automorphisms 1920 radius 2 diameter 2 girth ... regular graph Strongly regular br Hamiltonian graph Hamiltonian br Triangle free graph Triangle free br Cayley graph br Vertex transitive graph Vertex transitive br edge transitive graph Edge transitive br distance transitive graph Distance transitive . In the mathematics mathematical field of graph theory , the Clebsch graph ref name MathWorld ref Brouwer et al. 1989 use name Clebsch graph for a different, but related graph. ref is an undirected graph with 16 vertices and 40 edges. It is named ... Gleason graph . ref A. Clebsch, Ueber die Fl chen vierter Ordnung, welche eine ... math.cudenver.edu wcherowi courses m6023 shilpa.pdf The Clebsch Graph on Bill Cherowitzo s home page ref Construction This graph is equivalent to the order 5 folded cube graph . It may be constructed by adding edges between opposite pairs of vertices in a 4 dimensional hypercube graph. In an n dimensional ..., it can be formed from a 5 dimensional hypercube graph by Vertex identification identifying ... graph, is to create a vertex for each element of the finite field GF 16 , and connect two vertices ... potenza.ps page 6 ref Properties The Clebsch graph is a strongly regular graph of degree 5 with parameters ... ref Its complement is also a strongly regular graph. ref name MathWorld cite web url http mathworld.wolfram.com ClebschGraph.html title Clebsch Graph. last Weisstein first Eric W. publisher From MathWorld A Wolfram Web Resource accessdate 2009 08 13 ref ref name Cherowitzo The graph is Hamiltonian graph hamiltonian , Planar graph non planar and eulerian graph non eulerian . It is also both 5 k vertex connected graph vertex connected and 5 k edge connected graph edge connected . The induced subgraph subgraph that is induced by the ten non neighbors of any vertex in the Clebsch graph forms an graph ... more details
infobox graph name Errera graph image Image Errera graph alt.svg 220px image caption The Errera graph namesake Alfred Errera vertices 17 edges 45 automorphisms 20 dihedral group D sub 10 sub girth 3 radius 3 diameter 4 chromatic number 4 chromatic index 6 properties Planar graph Planar br Hamiltonian graph Hamiltonian ref MathWorld urlname HamiltonianGraph title Hamiltonian Graph ref In the mathematics mathematical field of graph theory , the Errera graph is a graph with 17 vertices and 45 edges discovered by Alfred Errera. ref MathWorld urlname ErreraGraph title Errera graph ref Published in 1921, it provides an example of how Alfred Kempe Kempe s proof of the four color theorem cannot work. ref Errera, A. Du coloriage des cartes et de quelques questions d analysis situs. Ph.D. thesis. 1921. ref ref Peter Heinig. http www m9.ma.tum.de foswiki pub Allgemeines PeterHeinig erreraGraphIsNarrowProof.pdf Proof that the Errera Graph is a narrow Kempe Impasse . 2007. ref Later, the Fritsch graph and Soifer graph provide two smaller counterexamples. ref Gethner, E. and Springer, W. M. II. How False Is Kempe s Proof of the Four Color Theorem? Congr. Numer. 164, 159 175, 2003. ref The Errera graph is Planar graph planar and has chromatic number 4, chromatic index 6, radius 3, diameter 4 and girth graph theory girth 3. All its vertices are of degree 5 or 6 and it is a 5 k vertex connected graph vertex connected graph and a 5 k edge connected graph edge connected graph . Algebraic properties The Errera graph is not a vertex transitive graph and its full automorphism group is isomorphic .... The characteristic polynomial of the Errera graph is math x 2 2 x 5 x 2 x 1 2 x 3 4 x 2 9 x 10 x 4 2 x 3 7 x 2 18 x 9 2 math . Gallery gallery Image Errera graph 4COL.svg The chromatic number of the Errera graph is 4. Image Errera graph 6color edge.svg The chromatic index of the Errera graph is 6. Image Errera graph.svg The Errera graph is Planar graph planar . gallery References ... more details
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