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Topic: Cayley graph

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Girton College

	Cayley Graphs
		The graph associated with the puzzle is then very regular, and is called the Cayley graph of the group.
		Most Cayley graphs however cannot be realised in two or three dimensions with all its edges the same length.
		Cayley graphs are uniform, so this dangling string net will look exactly the same regardless of which node you picked it up at.
	www.geocities.com /jaapsch/puzzles/cayley.htm (4449 words)

	Cayley graphs
		Cayley graphs are named for Arthur Cayley (August 1821-January 1895), who though starting out as a laywer, eventually published over 900 papers and notes covering nearly every aspect of modern mathematics.
		Moreover, a solution of the Rubik's cube is simply a path in the graph from the vertex associated to the present position of the cube to the vertex associated to the identity element.
		The number of moves in the shortest possible solution is simply the distance from the vertex associated to the present position of the cube to the vertex associated to the identity element.
	web.usna.navy.mil /~wdj/book/node186.html (303 words)

	Cubic Cages
		The graph is not vertex-transitive having orbits of length 8 and 16.
		Collapsing the triangles to a vertex yields a 2688-vertex graph with diameter 14 and girth 18.
		A smallest graph of girth 10 and valency 3.
	people.csse.uwa.edu.au /gordon/cages/index.html (2642 words)

	Dissertation Defense, WMU Graduate College
		Graphs, groups, and surfaces are all subjects of study in topological graph theory, using techniques and principles from the disciplines of graph theory, algebra, and topology.
		A Cayley graph provides a graphical representation of a finite group and a fixed generating set for the group; a Cayley map is a two?cell imbedding into a surface of a Cayley graph such that labeled outward?directed darts occur in the same sequence at each vertex.
		The set of all automorphisms of a Cayley map M form a group, Aut M. A Cayley map that achieves its maximum possible number of automorphisms is called a symmetrical Cayley map.
	www.wmich.edu /graduate/dissertation/dis-archive/smith.html (411 words)

	PlanetMath: Schützenberger graph
		Schützenberger graphs play in combinatorial inverse semigroups theory the role that Cayley graphs play in combinatorial group theory.
		Cross-references: identity, group, Cayley graphs, graphs, class, Wagner congruence, generated by, congruence, edge, vertex, word, Green relation, equivalence class, inverse semigroup, monoid, presentation
		This is version 31 of Schützenberger graph, born on 2006-08-20, modified 2006-09-26.
	planetmath.org /encyclopedia/SchutzenbergerGraph.html (168 words)

	Arbitrary length walks in Cayley graph
		The diameter of = the Cayley graph of S_n on the above generating set is n(n-1)/2 - the = longest element is the order-reversing permutation.
		More to the point, the functions f_j are paths from another Cayley graph, say C', with a different generating set that have been mapped to paths containing only the previously mentioned transposition generators.
		The diameter of the Cayley graph of S_n on the above generating set is n(n-1)/2 - the longest element is the order-reversing permutation.
	www.forum-one.org /new-3626223-4348.html (754 words)

	Automatic Groups
		In order to define word-hyperbolic groups, we need to discuss Cayley graphs; the Cayley graph is a graph with a point for each element of the group and with a directed line from one point to a second point whenever the second element is the first element times one of the generators.
		A path in a Cayley graph is a series of lines that match start to end; a path corresponds to multiplication by a series of generators.
		A word-hyperbolic group is one whose Cayley graph has the following property: given a triangle with edges of shortest possible length to still join the vertices, the distance from a point on one edge to the union of the other two edges is bounded by some constant.
	www.geom.uiuc.edu /docs/forum/automaticgroups/automaticgroups.html (1192 words)

	[No title]
		Thus such a graph really is a Cayley diagram of the group G and the generators x_i.
		So all the graphs you are interested in are really Cayley graphs in disguise _if_ you know that the stabilizer of one (hence any) point is trivial.
		This graph has 10 vertices; if it is a Cayley graph, then it is the Cayley graph of some group G with 10 elements.
	www.math.niu.edu /Papers/Rusin/known-math/96/transitive_graphs (1012 words)

	Dave Witte Morris' papers in graph theory
		For any Cayley graph on any finite abelian group, this paper determines precisely which elements of the cycle space can be written as sums of hamiltonian cycles.
		For any Cayley graph on any finite abelian group of odd order, this paper determines precisely which flows can be written as sums of hamiltonian cycles.
		This yields the first known construction of Cayley graphs for which 2g - g' is arbitrarily large, where g and g' are the orientable genus and the non-orientable genus of the Cayley graph.
	people.uleth.ca /~dave.morris/GraphTheory.shtml (1225 words)

	PlanetMath: Cayley graph
		That is, the vertices of the Cayley graph are precisely the elements of
		This is version 2 of Cayley graph, born on 2002-06-26, modified 2002-06-26.
		(Combinatorics :: Graph theory :: Graphs and groups)
	planetmath.org /encyclopedia/CayleyGraph.html (94 words)

	Graph (mathematics) - Wikipedia, the free encyclopedia
		A quiver is sometimes said to be simply a directed graph, but in practice it is a directed graph with vector spaces attached to the vertices and linear transformations attached to the arcs.
		In a weighted graph or digraph, each edge is associated with some value, variously called its cost, weight, length or other term depending on the application; such graphs arise in many contexts, for example in optimal routing problems such as the traveling salesman problem.
		Every graph gives rise to a matroid, but in general the graph cannot be recovered from its matroid, so matroids are not truly generalizations of graphs.
	en.wikipedia.org /wiki/Graph_(mathematics) (1782 words)

	What is Geometric Group Theory? (Site not responding. Last check: 2007-10-21)
		The Cayley graph of G is a graph whose vertices are precisely the elements of G, and whose edges are described by the rule that for each pair of elements x, y of G there is an edge labeled by the generator s
		Of course, the Cayley graph depends on the choice of generators.
		When the group G is infinite, its Cayley graph reflects large-scale geometric features which can have a profound effect on the algebra of G.
	www.math.mcgill.ca /wise/ggt/cayley.html (307 words)

	RR-2814 : The Arrowhead Torus : a Cayley Graph on the 6-valent Grid
		RR-2814 : The Arrowhead Torus : a Cayley Graph on the 6-valent Grid
		RR-2814 - The Arrowhead Torus : a Cayley Graph on the 6-valent Grid
		Abstract : The «arrowhead torus»; is a broadcast graph that we define on the {\sl 6}-valent grid as a Cayley graph.
	www.inria.fr /rrrt/rr-2814.html (500 words)

	Todd Mateer: Permutation Groups
		A Cayley graph is a way to organize the behavior of the interactions between members of a group.
		An edge is placed between two vertices when it is possible to go from one element to the other by multiplying one of the elements by one of the group's generators or the inverse of one of these generators.
		However, the substantial improvement in the Cayley graph diameter upper bound of R(2,5) from several thousand to 236 demonstrates the powerful role that using computational techniques can play in the study of Abstract Algebra.
	dimacs.rutgers.edu /REU/1996/sims.html (1682 words)

	Cayley graph - Wikipedia, the free encyclopedia
		In mathematics, a Cayley graph (also known as a Cayley colour graph and named after Arthur Cayley), is a graph that encodes the structure of a group.
		For example, the Cayley graph of the free group on two generators a and b is depicted above and to the right respectively.
		Insights into the structure of the group can be obtained by studying the adjacency matrix of the graph and in particular applying the theorems of spectral graph theory.
	en.wikipedia.org /wiki/Cayley_graph (620 words)

	Colloquium - 5 December - 2002 - Department of Mathematics - University of Montana (Site not responding. Last check: 2007-10-21)
		A graph has the Cayley Isomorphism property if whenever it can be isomorphically represented as both the Cayley graph X(G;S) and X(G;S'), there is an automorphism of the group G that takes S to S'.
		A group has the Cayley Isomorphism property if all Cayley graphs on that group have the Cayley isomorphism property.
		he Cayley Isomorphism problem is the question of which groups, and which graphs, have the Cayley Isomorphism property.
	www.umt.edu /math/colloq/fall02/120502.html (179 words)

	On the clique number of random Cayley graphs (Site not responding. Last check: 2007-10-21)
		Firstly, take A to be the set of quadratic residues mod N. The resulting Cayley graph is called a Paley sum graph, and it is conjectured to have many of the properties enjoyed by random graphs.
		This is really a paper in combinatorial number theory, because cliques of size k in C(A) correspond exactly to sets X for which the restricted sumset X +^ X lies in A. (The restricted sumset consists of all sums x + x' with x not equal to x').
		We believe that this statement, which may be paraphrased as "Cayley graphs can be good Ramsey graphs", is new.
	www.stats.bris.ac.uk /~mabjg/abstracts/4.html (474 words)

	regcay (Site not responding. Last check: 2007-10-21)
		A Cayley graph has vertex set the elements of a group G and edge set determined by a balanced generating set X.
		A Cayley map M is an embedding of a Cayley graph in an oriented surface such that the local rotation at each vertex is described by the same permutation p of X.
		A graphical regular representation of group G is a Cayley graph Gamma =C(G,X) whose full automorphism group Aut Gamma is isomorphic to G.
	www.emba.uvm.edu /~archdeac/newlist/regcay.htm (368 words)

	VEGA 0.5 Quick Reference Manual: Functions in GROUPS.M
		FastCayley[g,op] finds the Cayley graph of a group generated by g with operation op.
		The coefficient at t^q is the number of graphs on p vertices and q edges.
		Schreier[g,H,op] finds the Schreier coset graph of a group generated by g with operation op with respect to the subgroup H.
	vega.ijp.si /Htmldoc/usages/GROUPS.HTM (1853 words)

	autocay
		x^{-1} is in X whenever x is in X), and a cyclic permutation p on X, a Cayley map CM(G,X,p) is a 2-cell embedding of the Cayley graph C(G,X) into an orientable surface with the same local orientation p at every vertex.
		A map-automorphism A of a Cayley map M = CM(G,X,p) is an oriented-region-preserving permutation of the set of arcs of M.
		While all the previously studied and well understood balanced and antibalanced cases certainly satisfy these conditions, the conditions do not seem to disqualify the possibility of the existence of a regular Cayley map that is neither balanced nor antibalanced.
	www.emba.uvm.edu /~archdeac/newlist/autocay.htm (343 words)

	in theory
		Definite the Cayley graph $G=(A,E)$ where every element of $A$ is a vertex and where $(a,b)$ is an edge if $a-b\in S$.
		It is the Cayley graph of ${\mathbb Z}_N$ with generators $\{-1,1\}$.
		The hypercube is the Cayley graph of the group $({\mathbb Z}_2)^n$ with the generator set that contains all the vectors that have precisely one 1.
	in-theory.blogspot.com (4234 words)

	On the Cayley graph of a generic finitely presented group, G. N. Arzhantseva, P.-A. Cherix
		On the Cayley graph of a generic finitely presented group
		We prove that in a certain statistical sense the Cayley graph of almost every finitely presented group with $m\ge 2$ generators contains a subdivision of the complete graph on $l\le 2m+1$ vertices.
		In particular, this Cayley graph is non planar.
	projecteuclid.org /getRecord?id=euclid.bbms/1102689123 (176 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Date: 26 May 1999 20:43:43 GMT Newsgroups: sci.math Keywords: Cayley graph of symmetry group of icosahedron In article
		Actually, in one sense it is the symmetry group of the dodecahedron/ icosahedron.
		If you draw the (discrete) group of symmetries as a graph, where vertices are group elements and edges are generators, then the resulting lattice is that of a soccer ball.
	www.mat.niu.edu /~rusin/known-math/99/cayley_icos (175 words)

	Parameters of directed strongly regular graphs: Constructions
		[13]) This graph is obtained by looking at (point,line) flags, where (p,L)->(q,M) when the flags are distinct and q is on L.
		[4]) This graph is obtained by looking at (point,line) flags, where (p,L)->(q,M) when the flags are distinct and q is on L.
		Let m and q be positive integers, and assume that each prime power in the complete factorization of m is 1 mod q.
	homepages.cwi.nl /~aeb/math/dsrg/dsrg-2.html (1456 words)

	Graph TeX (Site not responding. Last check: 2007-10-21)
		If you are anal retentive like me, there are certain things you expect to see in a graph: smoothly curved edges, spade-like arrowheads which follow the curve of the edge, TeX labels, etc. Graph-TeX tries to meet these expectations by performing tedious calculations from high school trigonometry.
		If your graph can be drawn with all its vertices lying on a rectilinear grid, you needn't be aware of the Perl interface aside from carrying out the
		On the other hand, some graphs, such as the Petersen graph, are not naturally drawn with their vertices on any rectilinear grid.
	www.ima.umn.edu /~pliam/gtht/gtht.html (129 words)

	RR-2702 : The Diamond Torus : a Cayley Graph on the {\sl 6}-valent Grid
		Abstract : We pursue our analysis of a family of Cayley graphs defined on the { \sl 6}-valent grid from generators and relations and provided with a high level of symmetry.
		The «diamond torus»; is the graph of a finite group generated by superimposing a cyclic relation for each direction.
		As a Cayley graph, it allows recursive co nstructions and divide-and-conquer schemes for information dissemination, it is also vertex-transitive hence all routers!
	www.inria.fr /rrrt/rr-2702.html (524 words)

	Math Magic
		But there is another graph of a sliding puzzle called a Cayley graph.
		There are 12 different positions for this puzzle, and each has 2 possible moves, so the Cayley graph must be a 12-cycle.
		Joseph DeVincentis found that the Cayley graph of an n-cycle with (n-1) pieces is an n(n-1)-cycle.
	www.stetson.edu /~efriedma/mathmagic/0700.html (221 words)

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