Selfref For information about graphs on Wikipedia, see Wikipedia Graphs and charts . Wiktionary Graph may refer to A Information graphics graphic such as a line chart , Plot graphics plot , 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 Graphemics wiktionary graphy graphy suffix Latin for to write or draw Graf Graff disambiguation List of information graphics software Disambiguation de Graph es Grafo desambiguaci n eu Grafo argipena fr Graphe hu Gr f egy rtelm s t lap 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 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 unused data Hungarian Journal for Industrial Chemistry 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
An acyclic graph may refer to Directed acyclic graph , a directed graph without any directed cycles Forest graph theory , an undirected acyclic graph Polytree , a directed graph without any undirected cycles mathdab ... more details
In mathematics, a k ultrahomogeneous graph is a graph mathematics graph in which every graph isomorphism isomorphism between two of its induced subgraph s of at most k vertices can be extended to an graph automorphism automorphism of the whole graph. If a graph is 5 ultrahomogeneous, then it is ultrahomogeneous for every k . The only finite connected graphs of this type are complete graph s, Tur n graph s, 3 × 3 rook s graph s, and the 5 cycle graph cycle . There are only two connected graphs that are 4 ultrahomogeneous but not 5 ultrahomogeneous the Schl fli graph and its complement. The proof relies on the classification of finite simple groups . ref harvtxt Buczak 1980 harvtxt Cameron 1980 harvtxt Devillers 2002 . ref The infinite Rado graph is countably ultrahomogeneous. Notes reflist Category Graph theory ... 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
The terms lattice graph , mesh graph , or grid graph refer to a number of categories of graph mathematics graph s whose graph drawing 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 1,..., 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 1 and m 1 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 es Gr fico de celos a ... 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 combin stub ... more details
otheruses4 the 3 regular graph the graph associated with a Coxeter group Coxeter diagram infobox graph name Coxeter graph image Image Coxeter graph.svg 250px image caption The Coxeter graph namesake vertices ... 4 chromatic number 3 chromatic index 3 properties Symmetric graph Symmetric br distance regular graph Distance regular br distance transitive graph Distance transitive br Cubic graph Cubic br Hypohamiltonian graph Hypohamiltonian In the mathematics mathematical field of graph theory , the Coxeter graph is a 3 regular graph with 28 vertices and 42 edges. ref MathWorld urlname CoxeterGraph title Coxeter Graph ref All the cubic graph cubic distance regular graph s are known. ref Brouwer, A. E. Cohen ... graph is one of the 13 such graphs. Properties The Coxeter graph has chromatic number 3, chromatic index 3, radius 4, diameter 4 and girth graph theory girth 7. It is also a 3 k vertex connected graph vertex connected graph and a 3 k edge connected graph edge connected graph . The Coxeter graph is hypohamiltonian graph hypohamiltonian it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian. It has Crossing number graph theory rectilinear crossing number 11, and is the smallest cubic graph with that crossing number currently known, but an 11 crossing, 26 vertex graph may exist OEIS id A110507 . The Coxeter graph may be constructed from the smaller distance regular Heawood graph by constructing a vertex for each 6 cycle in the Heawood graph and an edge for each disjoint pair of 6 cycles. ref citation first Italo J. last Dejter title From the Coxeter graph to the Klein graph journal Journal of Graph Theory year 2011 doi 10.1002 jgt.20597 arxiv 1002.1960 . ref Algebraic properties The automorphism group of the Coxeter graph ... ref It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Coxeter graph is a symmetric graph . It has automorphisms that take any vertex to any other vertex and any ... 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
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 ... volume 30 year 1959 pages 781 798 jstor 2237417 . ref It is a strongly 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 have two distinct neighbors ... 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 ... graph, this surface is a torus in which each vertex is surrounded by six triangles. ref Andries Brouwer Brouwer, A. E. http www.win.tue.nl aeb drg graphs Shrikhande.html 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 Brouwer first1 A. E ... publisher Springer Verlag pages 104 105 and 136 year 1989 . ref The Graph automorphism automorphism ... 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 F26 graph image File F26A graph.svg 220px image caption The F26A graph is Hamiltonian. namesake vertices 26 edges 39 automorphisms 78 girth 6 diameter 5 radius 5 chromatic number 2 chromatic index 3 properties Cayley graph br Symmetric graph Symmetric br Cubic graph Cubic br Hamiltonian graph Hamiltonian ref name Mathworld In the mathematics mathematical field of graph theory , the F26A graph is a symmetric graph symmetric bipartite graph bipartite cubic graph with 26 vertices and 39 edges. ref name Mathworld MathWorld urlname CubicSymmetricGraph title Cubic Symmetric Graph ref It has chromatic number 2, chromatic index 3, graph diameter diameter 5, radius 5 and girth graph theory girth 6. ref name COND Conder, M. and Dobcs nyi, P. Trivalent Symmetric ... a 3 k vertex connected graph vertex connected and 3 k edge connected graph edge connected graph. The F26A graph is hamiltonian graph Hamiltonian and can be described by the LCF notation &minus 7, 7 sup 13 sup . Algebraic properties The graph automorphism automorphism group of the F26A graph ... ref It acts transitively on the vertices, on the edges, and on the arcs of the graph. Therefore the F26A graph is a symmetric graph though not distance transitive graph distance transitive . It has ... census , the F26A graph is the only cubic symmetric graph on 26 vertices. ref name COND It is also a Cayley graph for the dihedral group D sub 26 sub , generated by a , ab , and ab sup 4 ... Graphs , p. 67. ref math D 26 langle a, b a 2 b 13 1, aba b 1 rangle . math The F26A graph is the smallest cubic graph where the automorphism group regular group action acts regularly on arcs ... graph is equal to math x 3 x 3 x 4 5x 2 3 6. , math Gallery gallery Image F26A graph 2COL.svg The chromatic number of the F26A graph is 2. Image F26A graph 3color edge.svg The chromatic index of the F26A graph is 3. Image F26A graph alt.svg Alternative drawing of the F26A graph. gallery ... more details
In the mathematics mathematical field of graph theory , the null graph may refer either to the order graph theory order zero graph mathematics graph , or alternatively, to any edgeless graph the latter is sometimes called an empty graph . Order zero graph infobox graph name Order zero graph null graph ... index 0 genus 0 spectral gap undefined notation math K 0 math properties Integral graph Integral br Symmetric graph Symmetric The order graph theory order zero graph mathematics graph math K 0 math is the unique graph of order zero having zero vertex graph theory vertices . As a consequence, it also has zero edge graph theory edges . In some contexts, math K 0 math is excluded from being considered a graph either by definition, or more simply as a matter of convenience . The order zero graph ... of a category of graphs. Its inclusion within the definition of graph theory is more useful in some ... theory set theoretic definitions of a graph it is the ordered pair of empty set s , and in recursive ... . On the negative side, most well defined formulas for graph properties must include exceptions for math K 0 math if it is included as a graph counting all strongly connected component s of a graph would become counting all non null strongly connected components of a graph . Due to the undesirable aspects, it is usually assumed in literature that the term graph implies graph with at least one vertex unless context suggests otherwise. ref MathWorld urlname EmptyGraph title Empty Graph ref ref MathWorld urlname NullGraph title Null Graph ref When acknowledged, math K 0 math fulfills vacuous truth vacuously most of the same basic graph properties as math K 1 math the graph with one vertex and no edges it has a size graph theory size of zero, it is equal to its complement graph math bar K 0 math , it is a connected component graph theory connected component namely, math forall x isin V forall y isin V exists path x,y math , a forest graph theory forest , and a planar graph . It may be an undirected ... more details
Image 6n graf.svg thumb 250px An example graph, with the properties of being planar graph planar and being connectivity graph theory connected , and with order 6, size 7, Distance graph theory diameter 3, girth graph theory girth 3, connectivity graph theory vertex connectivity 1, and degree sequence 3, 3, 3, 2, 2, 1 In graph theory , a graph property or graph invariant is a property of graph mathematics graphs that depends only on the abstract structure, not on graph representations such as particular graph labeling labellings or graph drawing drawings of the graph. Definitions While graph drawing and graph representation are valid topics in graph theory, in order to focus only on the abstract structure of graphs, a graph property is defined to be a property preserved under all possible graph isomorphism isomorphism s of a graph. In other words, it is a property of the graph itself, not of a specific drawing or representation of the graph. Informally, the term graph invariant is used for properties ... of graphs. For example, the statement graph does not have vertices of degree 1 is a property while the number of vertices of degree 1 in a graph is an invariant . More formally, a graph property is a class of graphs, i.e. a function from graphs to T,F , and a graph invariant is a function from graphs to some other set, ref R. Diestel, Graph Theory , 3rd edition, Heidelberg Springer Verlag, 2005 ... graphs have the same value. A graph property is often called hereditary property ... Noga author link Noga Alon last2 Shapira first2 Asaf title Every monotone graph property is testable ... under graph union disjoint union . ref Peter Mihok 1999 Reducible properties and uniquely partitionable ... zaI8tSABMncyewDU9RyJM PPA213,M1 p. 214 ref The property of being planar graph planar is both hereditary and additive, for example, since a subgraph of a planar graph must be planar, and a disjoint union of two planar graphs must also be planar. The property of being connectivity graph theory connected ... 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 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 Multiple edges parallel 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 Levi graph image Image Pappus.png 240px image caption The Pappus graph , a Levi graph ... In combinatorics combinatorial mathematics , a Levi graph or incidence graph is a bipartite graph associated with an incidence structure . ref MathWorld urlname LeviGraph title Levi Graph ref ... a graph with one vertex per point, one vertex per line, and an edge for every incidence between ... Levi, F. W. title Finite geometrical systems year 1942 publisher Calcutta . ref The Levi graph of a system of points and lines usually has girth graph theory girth at least six Any 4 Cycle graph 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 may ... in Euclidean space . For every Levi graph, there is an equivalent hypergraph , and vice versa . Examples The Desargues graph is the Levi graph of the Desargues configuration , composed of 10 points ... graph can also be viewed as the generalized Petersen graph G 10,3 or the Kneser graph bipartite Kneser graph with parameters 5,2. It is 3 regular with 20 vertices. The Heawood graph is the Levi graph of the Fano plane . It is also known as the 3,6 cage graph theory cage , and is 3 regular with 14 vertices. The M bius Kantor graph is the Levi graph of the M bius Kantor configuration , a system ... regular with 16 vertices. The Pappus graph is the Levi graph of the Pappus configuration , composed ... passing through each point. It is 3 regular with 18 vertices. The Gray graph is the Levi graph of a configuration ... lines through them. The Tutte eight cage is the Levi graph of the Cremona Richmond configuration ... graph Q sub 4 sub is the Levi graph of the M bius configuration formed by the points and planes of two mutually incident tetrahedra. The Ljubljana graph on 112 vertices is the Levi graph of the Ljubljana ... Graph. 2002. http citeseer.ist.psu.edu conder02ljubljana.html . ref References reflist ... 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 Mathematical chess problems Category Parametric families of graphs ... more details
infobox graph name Frucht graph image File Frucht planar Lombardi.svg 200px image caption The Frucht graph namesake Robert Frucht vertices 12 edges 18 automorphisms 1 id girth 3 radius 3 diameter 4 chromatic number 3 chromatic index 3 properties Cubic graph Cubic br Planar graph Planar br Hamiltonian graph Hamiltonian In the mathematics mathematical field of graph theory , the Frucht graph is a 3 regular graph with 12 vertices, 18 edges, and no nontrivial graph automorphism symmetries . ref MathWorld urlname FruchtGraph title Frucht Graph ref It was first described by Robert Frucht in 1939. ref name f38 The Frucht graph is a Halin graph with chromatic number 3, chromatic index 3, radius 3, diameter 4 and Girth graph theory girth 3. As with every Halin graph, the Frucht graph is planar graph planar , 3 k vertex connected graph vertex connected , and polyhedral graph polyhedral . It is also a 3 k edge connected graph edge connected graph . The Frucht graph is Hamiltonian graph Hamiltonian ... graph is one of the two smallest cubic graphs possessing only a single graph automorphism , the identity ref Skiena, S. Implementing Discrete Mathematics Combinatorics and Graph Theory with Mathematica ... from every other vertex . Such graphs are called asymmetric graph asymmetric or identity graphs ... of a graph, ref name f38 Citation last1 Frucht first1 R. title Herstellung von Graphen mit vorgegebener ... of a 3 regular graph ref Citation last1 Frucht first1 R. title Graphs of degree three with a given ... Canadian Journal of Mathematics issn 0008 414X volume 1 pages 365 378 . ref the Frucht graph provides an example of this realization for the trivial group . The characteristic polynomial of a graph characteristic polynomial of the Frucht graph is math x 3 x 2 x x 1 x 2 x 3 x 2 2 x 1 x 4 x 3 6 x 2 5 x 4 math . Gallery gallery File Frucht graph 3COL.svg The chromatic number of the Frucht graph is 3. File Frucht Lombardi.svg The Frucht graph is Hamiltonian graph Hamiltonian . gallery See also ... 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 graph 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
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