Proofs of convergence of random variables - Revision history

en>Jonesey95: Fixing Category:CS1 errors: LCCN error

2014-09-14T02:48:35Z

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-{{infobox graph
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-The friendship graph ''F<sub>n</sub>'' can be constructed by joining ''n'' copies of the [[cycle graph]] ''C''<sub>3</sub> with a common vertex.<ref>Gallian, J. A. "Dynamic Survey DS6: Graph Labeling." Electronic J. Combinatorics, DS6, 1-58, Jan. 3, 2007. [http://www.combinatorics.org/Surveys/ds6.pdf].</ref>
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-By construction, the friendship graph ''F<sub>n</sub>'' is isomorphic to the [[windmill graph]] Wd(3,''n''). It is [[Unit distance graph|unit distance]] with girth 3, diameter 2 and radius 1. The graph ''F''<sub>2</sub> is isomorphic to the [[butterfly graph]].
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-==Friendship theorem==
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en>ChrisGualtieri: /* {{anchor|propA1i}} Convergence in probability does not imply almost sure convergence in the discrete case */General Fixes using AWB

2013-10-26T00:02:16Z

{{anchor|propA1i}} Convergence in probability does not imply almost sure convergence in the discrete case: General Fixes using AWB

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			{{infobox graph
			\| name = Friendship graph
			\| image = [[Image: Friendship graph 8.svg\|120px]]
			\| image_caption = The friendship graph ''F''<sub>8</sub>.
			\| vertices = ''2n+1''
			\| edges = ''3n''
			\| automorphisms =
			\| chromatic_number = 3
			\| girth = 3
			\| diameter = 2
			\| radius = 1
			\| chromatic_index = ''2n''
			\|notation = ''F<sub>n</sub>''
			\| properties = [[Unit distance graph\|Unit distance]]<br>[[planar graph\|Planar]]<br>[[Eulerian graph\|Eulerian]]<br>[[Factor-critical graph\|factor-critical]]
			}}
			[[Image:Friendship graphs.svg\|thumb\|450px\|The friendship graphs ''F''<sub>2</sub>, ''F''<sub>3</sub> and ''F''<sub>4</sub>.]]
			In the [[mathematics\|mathematical]] field of [[graph theory]], the '''friendship graph''' (or '''Dutch windmill graph''' or '''''n''-fan''') ''F''<sub>''n''</sub> is a [[planar graph\|planar]] [[undirected graph]] with ''2n+1'' vertices and ''3n'' edges.<ref>{{MathWorld\|urlname=DutchWindmillGraph\|title=Dutch Windmill Graph}}</ref>

			The friendship graph ''F<sub>n</sub>'' can be constructed by joining ''n'' copies of the [[cycle graph]] ''C''<sub>3</sub> with a common vertex.<ref>Gallian, J. A. "Dynamic Survey DS6: Graph Labeling." Electronic J. Combinatorics, DS6, 1-58, Jan. 3, 2007. [http://www.combinatorics.org/Surveys/ds6.pdf].</ref>

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			==Friendship theorem==
			The '''friendship theorem''' of {{harvs\|first1=Paul\|last1=Erdős\|author1-link=Paul Erdős\|first2=Alfréd\|last2=Rényi\|author2-link=Alfréd Rényi\|first3=Vera T.\|last3=Sós\|author3-link=Vera T. Sós\|year=1966\|txt}}<ref>{{citation\|first1=Paul\|last1=Erdős\|author1-link=Paul Erdős\|first2=Alfréd\|last2=Rényi\|author2-link=Alfréd Rényi\|first3=Vera T.\|last3=Sós\|author3-link=Vera T. Sós\|url=http://www.renyi.hu/~p_erdos/1966-06.pdf\|title=On a problem of graph theory\|journal=Studia Sci. Math. Hungar.\|volume=1\|year=1966\|pages=215–235}}.</ref> states that the finite graphs with the property that every two vertices have exactly one neighbor in common are exactly the friendship graphs. Informally, if a group of people has the property that every pair of people has exactly a friend in common, then there must be one person who is a friend to all the others. However, for infinite graphs, there can be many different graphs with the same cardinality that have this property.<ref>{{citation\|first1=Václav\|last1=Chvátal\|author1-link=Václav Chvátal\|first2=Anton\|last2=Kotzig\|first3=Ivo G.\|last3=Rosenberg\|first4=Roy O.\|last4=Davies\|title=There are <math>\scriptstyle 2^{\aleph_\alpha}</math> friendship graphs of cardinal <math>\scriptstyle\aleph_\alpha</math>\|journal=Canadian Mathematical Bulletin\|volume=19\|issue=4\|year=1976\|pages=431–433}}. </ref> A combinatorial proof was given by Mertzios and Unger. <ref>{{cite journal\|last=Mertzios\|first=George\|coauthors=Walter Unger\|title=The friendship problem on graphs\|journal=Relations, Orders and Graphs: Interaction with Computer Science\|date=2008\|url=http://www.dur.ac.uk/george.mertzios/papers/Conf/Conf_Windmills.pdf}}</ref> Another proof was given by Craig Huneke <ref>Huneke, Craig. [http://www.jstor.org/discover/10.2307/2695332?uid=3739832&uid=2&uid=4&uid=3739256&sid=21102015693863 "The Friendship Theorem."] The American Mathematical Monthly 109.2 (2002): 192-94. JSTOR. Mathematical Association of America. Web. <http://www.jstor.org/discover/10.2307/2695332?uid=3739832&uid=2129&uid=2&uid=70&uid=4&uid=3739256&sid=21102213900737></ref>


			==Labeling and colouring==
			The friendship graph has [[chromatic number]] 3 and [[chromatic index]] ''2n''. Its [[chromatic polynomial]] can be deduced from the chromatic polynomial of the cycle graph ''C''<sub>3</sub> and is equal to <math>(x-2)^n (x-1)^n x</math>.

			The friendship graph ''F<sub>n</sub>'' is [[Edge-graceful labeling\|edge-graceful]] if and only if ''n'' is odd. It is [[Graceful labeling\|graceful]] if and only if ''n'' ≡ 0 (mod 4) or ''n'' ≡ 1 (mod 4).<ref>J.C. Bermond, A. E. Brouwer, and A. Germa, "Systèmes de triplets et différences associées", Problèmes Combinatoires et Théorie des Graphes, Colloq. Intern. du CNRS, 260, Editions du Centre Nationale de la Recherche Scientifique, Paris (1978) 35-38.</ref><ref>J. C. Bermond, A. Kotzig, and J. Turgeon, On a combinatorial problem of antennas in radioastronomy, in Combinatorics, A. Hajnal and V. T. Sos, eds., Colloq. Math. Soc. János Bolyai, 18, 2 vols. North-Holland, Amsterdam (1978) 135-149.</ref>

			Every friendship graph is [[Factor-critical graph\|factor-critical]].

			==Extremal graph theory==
			According to [[extremal graph theory]], every graph with sufficiently many edges (relative to its number of vertices) must contain a ''k''-fan. More specifically, this is true for an ''n''-vertex graph if the number of edges is
			:<math>\left\lfloor \frac{n^2}{4}\right\rfloor + f(k),</math>
			where ''f''(''k'') is ''k''<sup>2</sup> − ''k'' if ''k'' is odd, and
			''f''(''k'') is ''k''<sup>2</sup> − 3''k''/2 if ''k'' is even. These bounds generalize [[Turán's theorem]] on the number of edges in a [[triangle-free graph]], and they are the best possible bounds for this problem, in that for any smaller number of edges there exist graphs that do not contain a ''k''-fan.<ref>{{citation
			\| last1 = Erdős \| first1 = P. \| author1-link = Paul Erdős
			\| last2 = Füredi \| first2 = Z. \| author2-link = Zoltán Füredi
			\| last3 = Gould \| first3 = R. J. \| author3-link = Ronald J. Gould
			\| last4 = Gunderson \| first4 = D. S. \| author4-link = David Shane Gunderson
			\| doi = 10.1006/jctb.1995.1026
			\| issue = 1
			\| journal = Journal of Combinatorial Theory
			\| mr = 1328293
			\| pages = 89–100
			\| series = Series B
			\| title = Extremal graphs for intersecting triangles
			\| url = http://www.math.uiuc.edu/~z-furedi/PUBS/furedi_erdos_gould_gunderson_triangles.ps
			\| volume = 64
			\| year = 1995}}.</ref>

			== References ==
			{{reflist}}

			[[Category:Parametric families of graphs]]
			[[Category:Planar graphs]]

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