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Topic: Symmetric graph

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	PlanetMath: directed graph
		Digraphs are generally drawn in a similar manner to graphs with arrows on the edges to indicate a sense of direction.
		Cross-references: arrows, graph, isomorphic, symmetric, arcs, edges, subset, vertices
		This is version 2 of directed graph, born on 2002-02-03, modified 2002-02-21.
	planetmath.org /encyclopedia/DirectedGraph.html (89 words)

	Algebra II: Quadratic Functions - Math for Morons Like Us (Site not responding. Last check: 2007-10-11)
		A graph is said to be symmetric to the y-axis when (x, y) and (-x, y) are points on the graph.
		A graph is said to be symmetric to the x-axis when (x, y) and (x, -y) are points on the graph.
		A graph is said to be symmetric to the origin when (x, y) and (-x, -y) are points on the graph.
	library.thinkquest.org /20991/alg2/quad2.html (568 words)

	Tom Sawyer Software: Image Gallery (Site not responding. Last check: 2007-10-11)
		This simply structured symmetric graph drawing is hiding a more complex structure under each of its nodes.
		Entire symmetric graph drawing after the hidden nodes and edges are revealed.
		A hierarchical graph drawing that uses simultaneous constraints to show which different colored nodes are connected to blue nodes.
	www.tomsawyer.com /gallery/gallery.php (867 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-11)
		The information we have is: for the function to be symmetric to the y-axis f(x) = f(-x); for the function to be symmetric to the x-axis f(x) = -f(x); and for something to be symmetric to the origin then f(-x) = -f(x).
		Well, if a graph is symmetric about something, all it means is that you can fold the graph in some way, and the different parts of the graph will overlap.
		Finally, a graph that is symmetric about the origin will overlap if you fold the graph along the x-axis and the y-axis.
	mathforum.org /library/drmath/view/54481.html (610 words)

	graph
		Formally, a graph is a set of vertices and a binary relation between vertices, adjacency.
		Moreover, a mathematical graph is not a comparison chart, nor a diagram with an x- and y-axis, nor a squiggly line on a stock report.
		GraphEd -- Graph Editor and Layout Program (C), graph manipulation (C++, C, Mathematica, and Pascal), build, traverse, top sort, etc. weighted, directed graphs (Java), JGraphT (Java) build, traverse, and display directed and undirected graphs, GEF - Graph Editing Framework (Java) a library to edit and display graphs.
	www.nist.gov /dads/HTML/graph.html (571 words)

	MATH 114
		Close to the mean, the graph is concave down; that is, it is curving down.
		For the graph of the Normal distribution, the points of inflection occur a distance that is exactly one standard deviation from the mean.
		Since the graph is symmetric, this means that area between the values μ-3σ and μ is 0.4985, and the area between the values μ and μ+3σ
	jan.ucc.nau.edu /~d-ctel/mat/mat114/transcript1.htm (504 words)

	SparkNotes: Special Graphs: Symmetry
		If a graph does not change when reflected over a line or rotated around a point, the graph is symmetric with respect to that line or point.
		If a graph can be reflected over a line without altering the graph, then that line is called the axis of symmetry.
		The following graph is symmetric with respect to the origin.
	www.sparknotes.com /math/algebra2/specialgraphs/section1.html (312 words)

	Graphs and Symmetry
		Visually we have that given a point P on the graph if we draw a line segment PQ through P and the origin such that the origin is the midpoint of PQ, then Q is also on the graph.
		To test algebraically if a graph is symmetric with respect to the y axis, we replace all the x's with -x and see if we get an equivalent expression.
		To test algebraically if a graph is symmetric with respect to the origin we replace both x and y with -x and -y and see if the result is equivalent to the original expression.
	www.ltcconline.net /greenl/courses/103a/functions/symm.htm (378 words)

	Internet Topology - 2. Graph theoretic terms (AB)
		Graphs are directed in the sense that AB being an edge does not imply that BA is also an edge.
		To symmetrize a graph is to add reverse edge BA for each edge AB that is in the graph.
		The outdegree of a node in a directed graph is the number of links that originate at the node (number of B's such that AB is an edge.) Indegree is the number of links that terminate at the node.
	www.caida.org /~broido/ipt/ipt.grterm.html (1135 words)

	MTH 111 College Algebra -- OSU Extended Campus - Oregon State University (Site not responding. Last check: 2007-10-11)
		Use the graph of a function to determine intervals where the graph is increasing or decreasing.
		The graph of an even function is symmetric with respect to the y-axis.
		The graph of an odd function is symmetric with respect to the origin.
	oregonstate.edu /instruct/dce/mth111_2/two/2.2.htm (167 words)

	Curve sketching (Site not responding. Last check: 2007-10-11)
		The steepness of the graph depends on the parameters a and b: the derivative of the function at the point x is 2ax + b.
		The graph of a function of two variables is a surface in three dimensions.
		We see that the graph of the function is a bowl with sides whose slopes increase as we move away from the center.
	www.chass.utoronto.ca /~osborne/MathTutorial/SKE.HTM (1127 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		A graph is symmetric with respect to the y-axis if whenever it is folded along the y-axis, both halves of the graph match.
		A graph is symmetric with respect to the x-axis if whenever it is folded along the x-axis, both halves match.
		A graph is symmetric with respect to the x-axis if whenever a point (a,b) lies on the graph, so does the point (a, -b).
	www2.latech.edu /~dehall/teams/algebra/definitions.htm (386 words)

	Direct interconnection networks I+II
		Graph G on Figure is edge symmetric, but it is not vertex symmetric.
		The n-dimensional binary hypercube (or n-cube) is a graph with 2
		Therefore, the neighborhood of a vertex in a SE digraph depends on its periodicity and the SE topology is not vertex symmetric.
	www.cs.wisc.edu /~tvrdik/5/html/Section5.html (4618 words)

	2001 Proposed REU Project Descriptions
		The idea of the project is to investigate cluster analysis from this graph theoretic vantage point, and to formulate cluster algorithms from the viewpoint of graph theoretic concepts such as connectivity, separation and bridges.
		A bridge of a symmetric graph is an edge whose removal causes the graph to have an extra connected component.
		Given an undirected graph G=(V,E) with n vertices, a natural number lambda is said to be a ``magic labeling'' if the edges of G can be labeled with nonnegative integers so that for each vertex the sum of the labels of the corresponding incident edges is lambda.
	dimacs.rutgers.edu /REU/2001/proposed.html (2093 words)

	Research
		By a geometric space I technically mean a complete edge colored graph: the graph is a set of n vertices with all n(n-1)/2 edges between them and the coloring is a function from the edges to their colors.
		Thus the four vertices of a rectangle give rise to an edge colored graph with four vertices and three interesting colors, one for the "horizontal" length, one for the "vertical" length and one for the "diagonal" length.
		A symmetric graph has a group of auotomorphisms that is transitive on the vertices and on the edges (of the one color) of the graph.
	employees.csbsju.edu /tsibley/research.htm (1377 words)

	Diffusion on aggregated graphs (AB)
		This is most likely due to the fact that a symmetric graph expands path counts in each and every node, whereas for directed graph paths generated in acyclic (destinations-bound) part cannot find continuation outside this part and are forced by unidirectional nature of this part to terminate in a small number of hops.
		Making graph changes the numer of nodes generating paths for common use, from 299 to 9449, whence an unexpected increase in path growth factor, which otherwise is just another nonlinear mean of nodes' outdegrees.
		More detailed information regarding the structure of the skitter AS graph is contained in the processing log of programs which prune trees or the whole acyclic part off the graph and find its giant component in the process of transitive closure.
	www.caida.org /~broido/bgp/diffus.html (1784 words)

	MTH 160 Homework Due Wednesday, March 29 (Site not responding. Last check: 2007-10-11)
		Neatly on graph paper plot the points and connect the dots to form a smooth neat version of the curve.
		You labeled three points on graph A. track where these three points went in the translation and label them in their new locations.
		You labeled three points on graph C. track where these three points went in the translation and label them in their new locations.
	users.stlcc.edu /amosher/EvenOddTranslationHW.htm (530 words)

	Math Games: Cubic Symmetric Graphs
		The graph is also symmetric -- any of the ten dominoes could be at the center (try it!).
		Foster Census, which is a listing of all of the cubic symmetric graphs up to 768 vertices.
		In a zero-symmetric graph, the vertices are equivalent, but the edges are maximally dissimilar.
	www.maa.org /editorial/mathgames/mathgames_12_29_03.html (982 words)

	On Symmetry in School Mathematics
		Hargittai and Hargittai in their text Symmetry: A Unifying Concept illustrate how deeply seated and ubiquitous symmetrical relationships are through hundreds of photographs of man-made objects, from examples in architectural symmetry, to those found in nature, as exemplified by the markings on the wings of a butterfly.
		So here we have a systematic (symmetric) way to represent an irrational number, that in and of itself is almost a contradiction of terms, for irrational numbers do not have systematic decimal representations; yet with the notions of continued fractions and infinite square roots there is a certain symmetry to them.
		Each of these graphs has its point of inflection on the y-axis, and the students easily prove that the graphs are symmetric with respect to their point of inflection.
	members.tripod.com /vismath/drei/index.html (2337 words)

	cylinder.html (Site not responding. Last check: 2007-10-11)
		The graph of the equation x + y = 4 is a straight line in two-space, and can be plotted parametrically by using x = s as the parameter, and writing y = 4 - x = 4 - s as follows:
		Now in three-space, the graph of x + y = 4 is a plane, and all we have seen so far is part of the graph in which z = 0.
		To obtain a more symmetric graph in all variables, you can use more symmetric parametrization.
	www.wfu.edu /~ekh/maple/cylinder/cylinder1.html (350 words)

	Polar Equations
		, the graph is a circle with a radius of 2.
		When k is even, the graph is symmetric with respect to both the x-axis and the y-axis.
		You see only one graph, because the negative values of a and b produce the same graph as the positives.
	jwilson.coe.uga.edu /emt668/EMAT6680.2001/Brown/Assignment11/polar.html (666 words)

	graphSol1
		is an even function and therefore its graph is symmetric around the y-axis.
		is neither even nor odd, so the graph is not symmetric about the y-axis or the origin.
		is undefined at -1, though, its graph does not cross the horizontal asymptote.
	www.math.ucdavis.edu /~marx/graphSol1/graphSol1.html (677 words)

	BioMath: Functions
		If you imagine a ball rolling on the graph, the ball will be rolling downhill on portions of the graph that decrease, and it will be rolling uphill on portions of the graph that increase.
		A graph is symmetric about the y-axis if the point (-x, y) lies on the graph whenever (x, y) does.
		The extrema depicted in the above graph are local because the function increases and decreases without bound as indicated by the arrows on the graph.
	www.biology.arizona.edu /BioMath/tutorials/Functions/Properties.html (880 words)

	Graphing Shortcuts
		y plane and sketch the graph by connecting the points, being careful not to connect two adjacent points unless all the input values lying between the two points are in the domain of the function.
		However there are certain shortcuts which may make the task of sketching graphs easier by reducing the number of points which must be plotted.
		A graph is symmetric with respect to the
	jwbales.home.mindspring.com /precal/part1/part1.7.html (866 words)

	Abstracts of papers by Peter J. Cameron
		We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince an interrogator with certainty that they have a colouring of the graph.
		In the finite case, all graphs on a given set of vertices are equivalent under switching, and we determine the structure of the switching group and show that its extension by the symmetric group on the vertex set is primitive.
		After an introduction to first-order logic and a discussion of which graph properties are first-order, we consider various logical concepts such as aleph_0-categoricity and homogeneity for graphs, and present a couple of theorems about finite graphs requiring logical techniques in their proofs.
	www.maths.qmw.ac.uk /~pjc/abstracts.html (9651 words)

	Symmetry and Graphing
		Because of this correspondence between the symmetry of the graph and the evenness or oddness of the function, "symmetry" in algebra is usually going to apply to the
		This graph (of a square-root function) shows no symmetry whatsoever, but it is a function.
		Graph C: This cubic is centered on the origin.
	www.purplemath.com /modules/symmetry3.htm (591 words)

	Mathwords: Symmetric with Respect to the Origin (Site not responding. Last check: 2007-10-11)
		Describes a graph that looks the same upside down or right side up.
		Formally, a graph is symmetric with respect to the origin if it is unchanged when reflected across both the x-axis and y-axis.
		Symmetric with respect to the x-axis, symmetric with respect to the y-axis, even function, odd function
	www.mathwords.com /s/symmetric_origin.htm (60 words)

	Sketching Graphs
		In this section, we'll demonstrate this by sketching the graphs of a few functions.
		We call this type of function odd : it implies that the graph is symmetric when rotated by 180 degrees about the origin.
		In terms of the graph, this means that it becomes very close to the axis.
	www.ugrad.math.ubc.ca /coursedoc/math100/notes/apps/graphs.html (437 words)

	Math Games: The Hoffman-Singleton Game
		He asked whether their were graphs which reached the upper bound.
		In addition, they discovered the fourth graph -- now called the Hoffman-Singleton graph -- one of the most remarkable objects in mathematics.Whether the last graph exists has been a famous unsolved problem since 1960.
		The remaining 30 cards are equivalent to one of the 4 (5,5) cage graphs.
	www.maa.org /editorial/mathgames/mathgames_11_01_04.html (1212 words)

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