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Topic: Path graph theory

	Path (graph theory) - Wikipedia, the free encyclopedia
		In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence.
		In modern graph theory, most often "simple" is implied; i.e., "cycle" means "simple cycle" and "path" means "simple path", but this convention is not always observed, especially in applied graph theory.
		In the graph shown, (1, 2, 5, 1, 2, 3) is a path of length 5, and (5, 2, 1) is a simple path of length 2.
	en.wikipedia.org /wiki/Path_(graph_theory) (401 words)

	Wikinfo \| Graph theory
		Graphs with weights can be used to represent many different concepts; for example if the graph represents a road network, the weights could represent the length of each road.The only information a weighted graph provides as such is (a) the vertices, (b) the edges and (c) the weights.
		Graph theory is also used to study molecules in chemistry and physics.
		Graph theory is the branch of mathematics that examines the properties of graphs.
	www.wikinfo.org /wiki.php?title=Graph_theory (2257 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.
		Unlike their Eulerian cousins, Hamiltonian paths and circuits can be hard to find, or even to tell whether they exist on a given graph G: it is known that finding a Hamiltonian path or circuit on a general graph G is an ``NP-complete'' problem.
	www.math.harvard.edu /~elkies/FS23j.04/glossary_graph.html (1317 words)

	Basic Graph Theory (Site not responding. Last check: 2007-10-13)
		A simple graph is a graph that does not have more than one edge between any two vertices and no edge starts and ends at the same vertex.
		A path is a sequence of vertices with the property that each vertex in the sequence is adjacent to the vertex next to it.
		The sum of the degrees of all the vertices of a graph is twice the number of edges in the graph.
	www.people.vcu.edu /~gasmerom/MAT131/graphs.html (354 words)

	Graph Theory
		Thus, the length of a path or cycle is also the number of edges in the path or cycle.
		In an undirected graph, this is obviously a metric.
		Bound δ (of a graph embedded in on a surface)
	www.math.fau.edu /locke/GRAPHTHE.HTM (1165 words)

	Ideas, Concepts, and Definitions (Site not responding. Last check: 2007-10-13)
		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
		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)

	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)

	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.
		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.
		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)

	Graphs
		The Graph Theory originates with a 1736 Euler's paper "The Seven Bridges of Königsberg".
		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.
		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 (1301 words)

	Graph Theory Glossary
		A circuit is a path which ends at the vertex it begins (so a loop is an circuit of length one).
		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.
	www.utm.edu /departments/math/graph/glossary.html (816 words)

	Graph Theory
		An acyclic graph (also known as a forest) is a graph with no cycles.
		There is a unique path between every pair of vertices in G. G is connected, and every edge in G is a bridge.
		On the other hand, if there is a spanning tree in G, there is a path between any pair of vertices in G; thus G is connected.
	www.personal.kent.edu /~rmuhamma/GraphTheory/MyGraphTheory/trees.htm (812 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.
		Exploration of Hamilton paths through a program which draws a snake that starts at one place in a box, and then extends itself until it cannot go further, after which it shrinks again, to seek another path.
	mathforum.org /library/topics/graph_theory (2440 words)

	Extremal graph theory (Site not responding. Last check: 2007-10-13)
		A property of a graph is monotone if the whole graph has the property when a subgraph does.
		A property of a graph is non-trivial if the empty graph does not have the property.
		The study of the minimum size of a graph with a monotone, non-trivial property, or the maximum size of a graph without it.
	john.fremlin.de /schoolwork/graph/graph-theory/node7.html (104 words)

	Graph Theory
		Explain that there are other types of graphs as well, and that some graphs can represent two types of graphs.
		Mention that a path can be of edges or vertices and the same applies to circuits.
		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)

	Amazon.com: Graph Theory: Books: Frank Harary (Site not responding. Last check: 2007-10-13)
		An effort has been made to present the various topics in the theory of graphs in a logical order, to indicate the historical background, and to clarify the exposition by including figures to illustrate concepts and results.
		It is no coincidence that graph theory has been independently discovered many times, since it may quite properly be regarded as an area of applied mathematics Read the first page
		If you are looking for examples of computer algorithms, look elsewhere; the closest this will get you is to "existence proofs", which is showing that something (such as a hamiltonian cycle) exists in a graph that has thus-and-such number of points or edges, but not tell you which sequence of points/edges make up that something.
	www.amazon.com /Graph-Theory-Frank-Harary/dp/0201410338 (1109 words)

	Graph Theory Lesson 12 (Site not responding. Last check: 2007-10-13)
		An Euler circuit on a graph G is a circuit that visits each vertex of G and uses every edge of G.
		An Euler path on a graph G is a path that visits each vertex of G and uses every edge of G.
		A graph that has directed edges is called a directed graph or sometimes just a digraph.
	oneweb.utc.edu /~Christopher-Mawata/petersen/lesson12.htm (469 words)

	Cycle (graph theory) - Wikipedia, the free encyclopedia
		A closed (simple) path, with no other repeated vertices than the starting and ending vertices.
		(This usage is common in graph theory.) This may also be called a simple cycle, circuit, circle, or polygon.
		(This usage is common in graph theory.) This may also be called a simple (directed) cycle.
	en.wikipedia.org /wiki/Cycle_(graph_theory) (260 words)

	Graph Theory
		Maximum flow through a network has many applications, both to practical situations and to theoretical results in graph theory.
		This module allows students the ability to move through a network to apply the max-flow algorithm, i.e., find augmenting paths, use the path to modify the flow and see the effect of that change.
		Users can either augment flow by clicking on vertices to extend paths, or they can test out their ability to find minimum cuts, or they can see the algorithm in action.
	www.conncoll.edu /CAT/projects/Nsf/graph/index.html (150 words)

	The Graph Theorists' Home Page Guide
		First of all, if you're a graph theorist or some person with strong interest in graph theory (you need not to be a mathematician!), and if you have a homepage but don't find a link to it on this page, please contact me as described above.
		PIGALE is a graph editor with an interface to the LEDA library and with many algorithms implemented essentially concerning planar graphs.
		"Graph Theory and Its Applications" (together with Jay Yellen), "a comprehensive applications-driven textbook that provides material for several different courses in graph theory." This site also provides links to other graph theoretical and mathematical resources.
	www.joergzuther.de /math/graph/homes.html (8736 words)

	Graph Theory - Euler (Site not responding. Last check: 2007-10-13)
		This problem has turned out to be fruitful since, in solving it, Euler invented graph theory, and (along with other accomplishments) played a vital role in the foundation of topology.
		An Euler path is a continuous path that traverses the paths (starting and ending on a vertex), passing over each path exactly once.
		More realistically, the natives simply hoped to find an Euler path (not necessarily a circuit) on which they did not have to cross the beautiful-once, but not-so-beautiful-the-second-time-around bridges only a single time, and ultimately cross just one or two bridges (for the second time) after following the completed path to get home again.
	members.aol.com /tylern7/math/euler-8.html (640 words)

	Briarskin: Graph Theory
		If a graph contains a twist point, then it does not contain a Hamiltonian cycle.
		Challenge: connect any number of nodes and edges to the terminal nodes of the above graph to form a graph with a Hamiltonian cycle.
		I noticed that if I drew a connection from B to C, then it would complete the two diagonals going from the top left to the lower right hand side of the matrix.
	www.geneffects.com /briarskin/theory/graph (250 words)

	GRAPH THEORY (Site not responding. Last check: 2007-10-13)
		This self-extracting program will allow you to experiment with undirected, unweighted graphs.
		It can also decide whether a graph is planar, connected, has an Euler circuit, Euler path, or Hamilton cycle.
		The graph can have up to 26 vertices.
	archives.math.utk.edu /software/msdos/discrete.math/graph/.html (49 words)

	The Problem of the Knight: A Fast and Simple Algorithm -- from Mathematica Information Center
		More than 200 years ago, Leonhard Euler posed the following problem: Given a chessboard of n times n squares, is it possible to find a path for the knight that touches every square exactly once in succession?
		For n >= 5, the answer is yes.
		Mathematica's list processing capabilities account for the compactness and readability of the resulting code.
	library.wolfram.com /infocenter/MathSource/909 (138 words)

	Graph Theory: Networking
		Home > Graph Theory: Networking > Shortest-Path Algorithms > Dijkstra's Algorithm
		Dijkstra's algorithm solves the single-source shortest-path problem on a weighted graph (directed or undirected), provided all edge-weights are nonnegative.
		Dijkstra's algorithm maintains a set of vertices whose final shortest-path weights from the source have already been calculated, along with a complementary set of vertices whose shortest-path weights have not yet been determined.
	links.math.rpi.edu /devmodules/graph_networking/compat/page13.html (178 words)

	Graph Theory: Networking
		Solve the single-pair shortest-path problem for the graph below and the pair
		What method did you use to find the shortest paths?
		Take a moment now to write down a method you would use to automate a search for a shortest path between any two vertices in an arbitrary graph.
	www.ibiblio.org /links/devmodules/graph_networking/xhtml/page12.xml (80 words)

	Introductory Graph Theory (Site not responding. Last check: 2007-10-13)
		The Konigsberg Bridge Problem: An Introduction to Eulerian Graphs
		A Traffic System Problem: An Introduction to Orientable Graphs
		The Three Houses and Three Utilities Problem: An Introduction to Planar Graphs
	www.wmich.edu /math-stat/people/faculty/chartrand/introgt (76 words)

	Graph Theory Tutorial (Site not responding. Last check: 2007-10-13)
		This is a very simple tutorial to let you know some terminology of Graph Theory using simple diagram of points and arcs.
		Check example application of graph theory in Q-Learning Tutorial
		See also: Q learning tutorial, Kardi Teknomo's Tutorial
	people.revoledu.com /kardi/tutorial/GraphTheory/index.html (39 words)

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