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Topic: Edge (graph theory)

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Bipartite graph

Max flow min cut theorem

	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 (Site not responding. Last check: 2007-10-13)
		Graph theory is the branch of mathematics that examines the properties of graphs.
		Depending on the applications edges may or not have a direction; edges joining a to itself may or may not be and vertices and/or edges may be assigned that is numbers.
		Every graph gives rise to a matroid but in general the graph cannot recovered from its matroid so matroids are truly generalizations of graphs.
	www.freeglossary.com /Graph_theory (1059 words)

	Edge - Biocrawler (Site not responding. Last check: 2007-10-13)
		In graph theory, a graph consists of a set of connections between objects.
		Edge is a colloquial adjective to describe escarpments in England e.g.:
		Edge Hill famous as the place of the first battle of the English Civil War.
	www.biocrawler.com /encyclopedia/Edge (392 words)

	14. Some Graph Theory
		Graphs are things that underlie many mathematical structures, and in fact anything that involves pairs of elements, and this includes any kind of relationship between pairs of individual entities.
		It is not an induced subgraph, since the edge (w,j) is an edge of path between vertices in this subgraph that is not an edge of the subgraph.
		A bridge is a set of edges or vertices and edges that that can be linked by paths of G not meeting C. The simplest kind of bridge is a chord of the cycle.
	www-math.mit.edu /18.310/some_graph_theory.html (3368 words)

	Graph Theory Glossary (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-10-13)
		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).
		A path is a sequence of consecutive edges in a graph and the length of the path is the number of edges traversed.
		Edges in graphs are undirected (as opposed to those in digraphs).
	www.utm.edu.cob-web.org:8888 /departments/math/graph/glossary.html (816 words)

	Graph (mathematics) - Wikipedia, the free encyclopedia
		An edge e = (x,y) is considered to be directed from x to y; y is called the head and x is called the tail of the edge; y is said to be a direct successor of x, and x is said to be a direct predecessor of y.
		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.
		A loop is an edge (directed or undirected) with both ends the same; these may be permitted or not permitted according to the application.
	en.wikipedia.org /wiki/Edge_(graph_theory) (1649 words)

	Graph theory glossary
		A coclique in a graph is a clique in its complementary graph (q.v.).
		girth (n.): The girth of a graph is the length of the shortest cycle(s) in the graph.
		When A,B are graphs, an isomorphism is a bijection from the vertices of A to the vertices of B such that any two vertices of A are adjacent if and only if their images in B are adjacent.
	www.math.harvard.edu /~elkies/FS23j.04/glossary_graph.html (1317 words)

	Intro to Graph Theory
		A subgraph of a graph is a subset of its points together with all the lines connecting members of the subset.
		A bridge is an edge whose removal from a graph increases the number of components (disconnects the graph).
		A local bridge of degree k is an edge whose removal causes the distance between the endpoints of the edge to be at least k.
	www.analytictech.com /networks/graphtheory.htm (1221 words)

	A Graph Theory Niche
		His graph proved that if there are more than two points (his graph had four points) with an odd number of lines to or from, it was not possible to cross all seven bridges just once.
		Graph theory was entirely new to Brigham and Dutton when they started studying it in the early 1980s.
		He is also using graph theory to find the shape of a large molecule consisting of hundreds of thousands of atoms, which has many important applications in the design of pharmaceuticals.
	www.cs.ucf.edu /newsletter/vol1/issue_two/graph-theory.html (1629 words)

	PlanetMath: graph
		A graph is then simple if there is at most one edge joining each pair of nodes.
		See Also: loop, neighborhood (of a vertex), Euler's polyhedron theorem, digraph, tree, spanning tree, connected graph, cycle, graph theory, graph topology, subgraph, simple path, Euler path, diameter, distance (in a graph), graph homomorphism, pseudograph, multigraph, order (of a graph)
		This is version 28 of graph, born on 2001-11-12, modified 2006-09-24.
	planetmath.org /encyclopedia/Graph.html (198 words)

	Graph Theory
		Example: This graph is not simple because it has 2 edges between the vertices A and B. Two vertices, v and w, of a graph are ADJACENT if there is an edge, vx, joining theem; the vertices are then considered INCIDENT to the edge, vx.
		Since one edge is incident with 2 vertices (note that G is simple), we can easily see that 1 handshake consists of 2 people, that is, 2 hands.
		In terms of graph theory, in any graph the sum of all the vertex-degrees is an even number - in fact, twice the number of edges.
	jwilson.coe.uga.edu /EMAT6680/Yamaguchi/emat6690/essay1/GT.html (1210 words)

	Graph Theory
		Two vertices of a graph that are connected by more than one edge are said to contain parallel edges or multiple edges. A graph with multiple edges is called a multigraph.
		is an edge connecting a vertex to itself or is incident on a single vertex.
		A finite graph G is always a finite union of maximal connected graphs which are called connected components of G.
	www.lv.psu.edu /ojj/courses/ist-230/topics/graphs.html (1649 words)

	Graph Theory
		In an undirected graph, this is obviously a metric.
		For a set S of edges, we use G[S] to denote the edge induced subgraph of G whose edge set is S and whose vertex set is the subset of V(G) consisting of those vertices incident with any edge in S.
		Bound δ (of a graph embedded in on a surface)
	www.math.fau.edu /locke/GRAPHTHE.HTM (1165 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)

	Glossary of graph theory - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-10-13)
		Graph theory is a growth area in mathematical research, and has a large specialized vocabulary.
		Likewise, a graph G is said to be homomorphic to a graph H if there is a mapping, called a homomorphism, from V(G) to V(H) such that if two vertices are adjacent in G then their corresponding vertices are adjacent in H.
		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.
	en.wikipedia.org.cob-web.org:8888 /wiki/Glossary_of_graph_theory (5923 words)

	``Introduction to Graph Theory'' (2nd edition)
		On a separate page is a discussion of the notation for the number of vertices and the number of edges of a graph G, based on feedback from the discrete mathematics community.
		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 (1095 words)

	Graph Theory
		An acyclic graph (also known as a forest) is a graph with no cycles.
		G is acyclic, and whenever any two arbitrary nonadjacent vertices in G are joined by and edge, the resulting enlarged graph G' has a unique cycle.
		Delete edges from G that are not bridges until we get a connected subgraph H in which each edge is a bridge.
	www.personal.kent.edu /~rmuhamma/GraphTheory/MyGraphTheory/trees.htm (812 words)

	Graph Theory
		A graph is a bunch of vertices and edges (also known as nodes and arcs).
		That's all a graph is. (We don't think of the vertices and edges as being located anywhere in space; a graph is completely specified once you've said that there are N vertices and M edges and these ones are joined to those ones.
		When such variations are allowed, the term "simple graph" is used for a graph without such features; when such variations are not allowed, one could specifically refer to a "multigraph with loops" for a graph that might have them.
	c2.com /cgi/wiki?GraphTheory (608 words)

	Graph Theory
		So the emphasis for the final will be on using graph theory as a tool to formulate problems, asking only for you to be familiar with a reasonable proportion of the material we've covered in class, including at least one of the class presentations in addition to that of your own group.
		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 (3531 words)

	Tom's Combinatorial Geometry Class
		Since every graph is determined by it's set of vertices and edges, we usually denote a graph by G=< V,E>.
		A graph is called a planar graph if it is drawn in such a way that the edges never cross, except at where they meet at vertices.
		There is a direct connection between polyhedra and planar graphs, namely that we can take any polyhedron and "project" it down onto a flat piece of paper, turning it into a graph.
	www.merrimack.edu /~thull/combgeom/graphnotes.html (1055 words)

	Edge and Adjacent Graph
		A graph that is not connected, of course it is called a disconnected graph.
		From these two vertices, an arc line was born, and the name of the arc is an edge.
		However, an edge cannot be supported by more than two vertices.
	people.revoledu.com /kardi/tutorial/GraphTheory/edge.html (101 words)

	Covering (graph theory) - Wikipedia, the free encyclopedia
		In the mathematical discipline of graph theory a covering for a graph is a set of vertices (or edges) so that the elements of the set are close (adjacent) to all edges (or vertices) of the graph.
		A vertex covering for a graph G is a set of vertices V so that every edge of G is incident to at least one vertex in V.
		An edge covering for a graph G is a set of edges E so that every vertex of G is adjacent to at least one edge in E.
	en.wikipedia.org /wiki/Covering_(graph_theory) (283 words)

	Amazon.com: Algebraic Graph Theory: Books: Chris Godsil,Gordon Royle (Site not responding. Last check: 2007-10-13)
		This book is primarily aimed at graduate students and researchers in graph theory, combinatorics, or discrete mathematics in general.
		However, all the necessary graph theory is developed from scratch, so the only pre-requisite for reading it is a first course in linear algebra and a small amount of elementary group theory.
		--It is sketchy on chromatic polynomial, planar graph.
	www.amazon.com /Algebraic-Graph-Theory-Chris-Godsil/dp/0387952209 (901 words)

	The Math Forum - Math Library - Graph Theory (Site not responding. Last check: 2007-10-13)
		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.
		A series of short interactive tutorials introducing the basic concepts of graph theory, designed with the needs of future high school teachers in mind and currently being used in math courses at the University of Tennessee at Martin.
		While all vertices and edges of the graph are similar, there are no edge-reversing automorphisms.
	mathforum.org /library/topics/graph_theory (2440 words)

	Unsolved Problems
		Let f(n) be the maximum possible number of edges in a graph on n vertices in which no two cycles have the same length.
		An (m,n)-cage is an m-regular graph with girth n and, subject to this, with the least possible number of vertices.
		The bandwidth of a graph G is the minimum bandwidth among adjacency matrices of graphs isomorphic to G.
	www.math.fau.edu /locke/Unsolved.htm (2911 words)

	Wiley::Graph Theory
		Intended neither to be a comprehensive overview nor an encyclopedic reference, this focused treatment goes deeply enough into a sufficiently wide variety of topics to illustrate the flavor, elegance, and power of graph theory.
		These strands center, respectively, around matching theory; planar graphs and hamiltonian cycles; topics involving chordal graphs and oriented graphs that naturally emerge from recent developments in the theory of graphic sequences; and an edge coloring strand that embraces both Ramsey theory and a self-contained introduction to Pólya's enumeration of nonisomorphic graphs.
		In the edge coloring strand, the reader is presumed to be familiar with the disjoint cycle factorization of a permutation.
	www.wiley.com /WileyCDA/WileyTitle/productCd-0471389250.html (375 words)

	Graph Theory
		Explain that there are other types of graphs as well, and that some graphs can represent two types of graphs.
		Assign groups of students to one of the graphs on the sheet.
		Students should find a path through all the edges, a path through all the vertices, a circuit through all the edges, and a circuit through all the vertices.
	www.shodor.org /succeed/mathcon/graphTheory.html (748 words)

	Graph Theory Lessons (Site not responding. Last check: 2007-10-13)
		The large one is for pictures of graphs and the lower one will have instructions and messages.
		Complete graphs are graphs in which every vertex is adjacent to every other vertex.
		A complete bipartite graph has the vertices in two groups (let us call them the left and right)and all vertices in one group are adjacent to all vertices in the other.
	oneweb.utc.edu /~Christopher-Mawata/petersen/lesson1.htm (615 words)

	Graph Theory: Industrial Drilling
		Now that you understand what a graph is, let's define some useful terms relative to the edges and vertices of a graph:
		An isolated vertex is a vertex of degree
		A graph is fully defined by its vertex and edge sets.
	www.ibiblio.org /links/devmodules/graph_theory/xhtml/page7.xml (120 words)

	Edge colouring and Vizing's theorem
		For the upper bound we do induction on the number of edges.
		Then we wish to recolour so all the edges are coloured.
		Certainly not all the colours were used to colour the new edges.
	john.fremlin.de /schoolwork/graph/graph-theory/node12.html (164 words)

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