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Topic: Vertex graph theory

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Edge (graph theory)

Bipartite graph

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	Degree (graph theory) - Wikipedia, the free encyclopedia
		In the mathematical field of graph theory the degree or valency of a vertex v is the number of edges incident to v (with loops being counted twice).
		In a directed graph the indegree of a vertex v is the number of edges terminating at v and the outdegree is the number of edges originating at v.
		If each vertex of the graph has the same degree k the graph is called a k-regular graph and the graph itself is said to have degree k.
	en.wikipedia.org /wiki/Degree_(graph_theory) (212 words)

	Graph theory - Wikipedia (Site not responding. Last check: 2007-09-07)
		Graph theory is the branch of mathematics that examines the properties of graphs.
		In computers, a finite directed or undirected graph (with n vertices, say) is often represented by its adjacency matrix: an n-by-n matrix whose entry in row i and column j gives the number of edges from the i-th to the j-th vertex.
		A subgraph of the graph G is a graph whose vertex set is a subset of the vertex set of G, whose edge set is a subset of the edge set of G, and such that the map w is the restriction of the map from G.
	wikipedia.findthelinks.com /gr/Graph_theory.html (1641 words)

	Graph theory - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-09-07)
		A graph structure can be extended by assigning a weight to each edge, or by making the edges to the graph directional (A links to B, but B does not necessarily link to A, as in webpages), technically called a digraph.
		Graphs are represented graphically by drawing a dot for every vertex, and drawing an arc between two vertices if they are connected by an edge.
		Incidence matrix - The graph is represented by a matrix of E (edges) by V (vertices), where contains the edge's data (simplest case: 1 - connected, 0 - not connected).
	encyclopedia.worldsearch.com /graph_theory.htm (968 words)

	PlanetMath: graph theory
		Graph theory is the branch of mathematics that concerns itself with graphs.
		It is usually agreed upon that graph theory proper was born in 1736, when Euler formalized the now-famous ``bridges of Königsberg'' problem.
		Now, a (finite) graph is usually thought of as a subset of pairs of elements of a finite set (called vertices), or more generally as a family of arbitrary sets in the case of hypergraphs.
	planetmath.org /encyclopedia/GraphTheory.html (506 words)

	Graph theory - Open Encyclopedia (Site not responding. Last check: 2007-09-07)
		Informally, a graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs).
		Depending on the applications, edges may or may not have a direction; edges joining a vertex to itself may or may not be allowed, and vertices and/or edges may be assigned weights, that is, numbers.
		If the vertex is an endpoint to the edge, a value of 1 is assigned to their crossing, otherwise, a value of 0 is assigned.
	open-encyclopedia.com /Graph_theory (881 words)

	Boost Graph Library: Graph Theory Review
		This chapter is meant as a refresher on elementary graph theory.
		Fundamentally, a graph consists of a set of vertices, and a set of edges, where an edge is something that connects two vertices in the graph.
		Vertex A is the source vertex and H is the target vertex.
	www.boost.org /libs/graph/doc/graph_theory_review.html (2374 words)

	Graphs
		The second notion, that of the edges being connections between nodes, is by far too important to the Graph Theory to leave it to one's intuitive perception.
		A degree of a vertex is the number of edges incident to it (loops being counted twice).
		For a graph, the sum of degrees of all its nodes equals twice the number of edges.
	www.cut-the-knot.org /do_you_know/graphs.shtml (1002 words)

	Graph theory
		An undirected graph G consist of a set of vertices,or nodes,V and a set of edges, or arcs, E such that each edge e is associated with an unordered pair of vertices.Thus
		A complete graph is a simple graph with n vertices in which there is an edge between every pair of distinct vertices.
		A graph G =(V, E) is bipartite if there exist subsets V1 and V2 (either possibly empty)of V such that V1 intersect V2 = empty set V1 union V2 = V,and each edge in E is incident on one vertex in V1 and one vertex in V2.
	www.nova.edu /~desir/graph.html (297 words)

	Graph Theory Open Problems
		A graph which can be embedded in the plane so that vertices correspond to points in the plane and edges correspond to unit-length line segments is called a ``unit-distance graph.'' The question above is equivalent to asking what the chromatic number of unit-distance graphs can be.
		Paul O'Donnell has found a unit distance graph of girth 12 which cannot be 3-colored, but this graph has an incredibly large number of points.
		To get the square of an oriented graph (or any directed graph) you leave the vertex set the same, keep all the arcs, and for each pair of arcs of the form (u,v), (v,w), you add the arc (u,w) if that arc was not already present.
	dimacs.rutgers.edu /~hochberg/undopen/graphtheory/graphtheory.html (705 words)

	05C: Graph theory
		A graph is a set V of vertices and a set E of edges -- pairs of elements of V. This simple definition makes Graph Theory the appropriate language for discussing (binary) relations on sets, which is clearly a broad topic.
		A graph may be viewed as a one-dimensional CW-complex and hence studied with tools from Algebraic Topology, in particular, questions of planarity (and genus).
		Determining the genus of a graph is NP-complete.
	www.math.niu.edu /~rusin/known-math/index/05CXX.html (1204 words)

	Ideas, Concepts, and Definitions (Site not responding. Last check: 2007-09-07)
		Graph paper is not particularly useful for drawing the graphs of Graph Theory.
		In Graph Theory, a graph is a collection of dots that may or may not be connected to each other by lines.
		If you look at a graph and your eyes want to zip all around it like a car on a race course, or if you notice shapes and patterns inside other shapes and patterns, then you are looking at the graph the way a graph theorist does.
	www.c3.lanl.gov /mega-math/gloss/graph/gr.html (215 words)

	Graph Theory
		Graph Theory was born to study problems of this type.
		The degree, d(v), of a vertex v is the number of edges with which it is incident.
		In an undirected graph, this is obviously a metric.
	www.math.fau.edu /locke/GRAPHTHE.HTM (1173 words)

	Graph Theory Glossary
		In a digraph (directed graph) the degree is usually divided into the in-degree and the out-degree (whose sum is the degree of the vertex in the underlying undirected graph).
		The vertex a is the initial vertex of the edge and b the terminal vertex.
		A path is a sequence of consecutive edges in a graph and the length of the path is the number of edges traversed.
	www.utm.edu /departments/math/graph/glossary.html (816 words)

	``Introduction to Graph Theory'' (2nd edition)
		"Even graph" is my compromise expression for the condition that all vertex degrees are even, and I will continue to use "cycle" for a 2-regular connected graph, "circuit" for a cyclically-edge-ordered connected even graph, and "circuit" for a minimal dependent set in a matroid.
		Most research and applications in graph theory concern graphs without multiple edges or loops, and often multiple edges can be modeled by edge weights.
		Letting "graph" forbid loops and multiple edges simplifies the first notion for students, making it possible to correctly view the edge set as a set of vertex pairs and avoid the technicalities of an incidence relation in the first definition.
	www.math.uiuc.edu /~west/igt (1028 words)

	Graph Theory
		The text is "Introduction to Graph Theory" by Richard J. Trudeau, which is in paperback from Dover Publications, NY, 1994; still in print and available in the bookstore or from amazon.com - here is a picture.
		Every graph G has embeddings in surfaces S_k when k is large enough (e.g., k = number of edges) and so there is a _least_ such nonnegative integer k which is defined to be the _genus of G_ denoted gamma(G) (the book uses g for genus but we use g for girth).
		The radius of a graph is the minimum eccentricity of the vertices, while the diameter of a graph is the maximum eccentricity of the vertices.
	www.georgetown.edu /faculty/kainen/graphtheory.html (2549 words)

	Graph Theory
		An acyclic graph (also known as a forest) is a graph with no cycles.
		A graph is connected if and only if it has a spanning tree.
		Repeat the process until we obtain either a single vertex (the center) or two vertices joined by an edge (the bicenter).
	www.personal.kent.edu /~rmuhamma/GraphTheory/MyGraphTheory/trees.htm (812 words)

	Vertex-transitive graph -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-07)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, a vertex-transitive (A drawing illustrating the relations between certain quantities plotted with reference to a set of axes) graph is a graph G such that, given any two vertices v
		In other words, a graph is vertex-transitive if its automorphism group acts (Click link for more info and facts about transitively) transitively upon its vertices.
		Every vertex-transitive graph is (A dependable follower (especially in party politics)) regular.
	www.absoluteastronomy.com /encyclopedia/v/ve/vertex-transitive_graph.htm (121 words)

	Graph Theory. 2. Vertex Descriptors and Graph Coloring
		For every sequence matrix three ordering criteria are applied: lexicographic ordering, based on strings of numbers, corresponding to every vertex, extracted as rows from sequence matrices; ordering by the sum of path lengths from a given vertex; and ordering by the sum of paths, starting from a given vertex.
		A sequence matrix is a matrix that collects vertex contribution in each row and elongation of paths in each column.
		The elements of the matrix cumulate a value of vertex property for a given elongation.
	lejpt.academicdirect.org /A01/37_52.htm (1115 words)

	Graph theory (Site not responding. Last check: 2007-09-07)
		The example graph is planar; the complete graph on n vertices, for n> 4, isn't planar.
		The example graph doesn't contain an Eulerian path, but it does contain a Hamiltonian path.
		Enthusiastic missionaries rejoiced at the thought.html">thought of a multitude the background.
	www.city-search.org /gr/graph-theory.html (1721 words)

	Graph Theory Spring 2005 (Site not responding. Last check: 2007-09-07)
		That means that for each vertex in the set {v_2, v_3,..., v_n} adjacent to v_1, there exists a vertex in {v_1, v_2,...,v_{n-1}} not adjacent to v_n.
		Check out the graph Tutte came up with to show that not all planar cubic graphs are Hamiltonian (as well as why this is both related to Tait and important).
		Graph theory was in an interview with Don Knuth this morning: listen here.
	cerebro.xu.edu /~smbelcas/graphthy.html (3258 words)

	Graph Theory (Site not responding. Last check: 2007-09-07)
		In practice one rarely has to contend with general graphs; often information is available to limit the graphs to a restricted class.
		Sometimes graph parameters that are not generally equal are equal for certain restricted classes.
		The objective is to partition the vertices of a graph into two sets S and T such that each vertex of one set is adjacent to a vertex in the other set, and the two sets are within one of having the same size.
	www.cs.usask.ca /projects/graph/index.shtml (420 words)

	Vertex Graph
		A graph is a set of points and arc.
		It is a story about the points and arc.
		In the beginning, in the family name Graph, there is a point.
	www.people.revoledu.com /kardi/tutorial/GraphTheory/vertex.html (30 words)

	Graph Theory (Site not responding. Last check: 2007-09-07)
		This paper develops a new method for visualizing graphs.
		The graph can further be viewed as a table of vertices and edges, where we read down the colmuns to find that, for example, edge A connects vertices 1 and 2:
		The other edges that contact either of verticex 1 or 9 are B, C and L, hence in the new graph, A is connected to B, C and L: This can also be placed in an adjacency matrix.
	www.users.globalnet.co.uk /~perry/maths/graphtheory/graphtheory.htm (137 words)

	Bipartite graph (Site not responding. Last check: 2007-09-07)
		In the mathematics field of graph theory, a bipartite graph is a special graph (mathematics) where the set of vertex (graph theory) can be divided into two disjunct sets with two vertices of the same set never sharing an edge (graph theory).
		Bipartite graphs are useful for modelling matching problems.
		We can model this as a graph with P + J the set of vertices.
	read-and-go.hopto.org /Graphs/Bipartite-graph.html (144 words)

	An overview on Graph Theory (Site not responding. Last check: 2007-09-07)
		A graph is a very simple structure consisting of a set of vertices and a family of lines (possibly oriented), called edges (undirected) or arcs (directed), each of them linking some pair of vertices.
		The number of concepts that can be defined on graphs is very large, and many generate deep problems or famous conjectures (for instance the four colour problem).
		We present in an annex a small bibliographical reference on Graph Theory, and a more precise description of our research topics.
	www-leibniz.imag.fr /GRAPH/english/overview.html (332 words)

	Graph Theory Tutorials (Site not responding. Last check: 2007-09-07)
		This is the home page for a series of short interactive tutorials introducing the basic concepts of graph theory.
		Starting with three motivating problems, this tutorial introduces the definition of graph along with the related terms: vertex (or node), edge (or arc), loop, degree, adjacent, path, circuit, planar, connected and component.
		This question can be changed to "how many colors does it take to color a planar graph?" In this tutorial we explain how to change the map to a graph and then how to answer the question for a graph.
	www.utm.edu /departments/math/graph (282 words)

	Complete graph (Site not responding. Last check: 2007-09-07)
		In the mathematics field of graph theory a complete graph is a simple graph where an edge (graph theory) connects every pair of vertex (graph theory).
		It is a regular graph of degree (graph theory)
		All complete graphs are their own Clique (graph theory)s.
	read-and-go.hopto.org /Graphs/Complete-graph.html (44 words)

	Open Directory - Science: Math: Combinatorics: Graph Theory (Site not responding. Last check: 2007-09-07)
		A Constructive Approach to Graph Theory - Notes on a semiotic approach to constructing isomorphism invariants of graphs by John-Tagore Tevet.
		Getgrats: General Theory of Graph Transformation Systems - A research network funded by the European Commission.
		Graph Colorings with Local Constraints - A survey by Zsolt Tuza.
	dmoz.org /Science/Math/Combinatorics/Graph_Theory (459 words)

	A compendium of NP optimization problems
		Graph Theory: Covering and Partitioning, Subgraphs and Supergraphs, Sets and Partitions.
		Graph Theory: Vertex Ordering, Network Design: Cuts and Connectivity.
		This is a continuously updated catalog of approximability results for NP optimization problems.
	www.nada.kth.se /~viggo/wwwcompendium (82 words)

	Graph Theory (Site not responding. Last check: 2007-09-07)
		A coloring of a simple graph is the assignment of a color to each vertex of the graph so that no two adjacent vertices are assigned the same color.
		Color each graph with as many colors as you can.
		Finally try to color the graph with the least number of colors possible.
	www.geom.uiuc.edu /~zarembe/graph1.html (138 words)

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