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| {{Graph families defined by their automorphisms}}
| | My name is Joann (34 years old) and my hobbies are Fantasy Football and Amateur radio.<br><br>my website: [http://Www.Dgzhenwang.com/plus/guestbook.php fifa 15 coin hack] |
| In [[graph theory]], a '''regular graph''' is a [[graph (mathematics)|graph]] where each vertex has the same number of neighbors; i.e. every vertex has the same [[Degree (graph theory)|degree]] or valency. A regular [[directed graph]] must also satisfy the stronger condition that the [[indegree]] and [[outdegree]] of each vertex are equal to each other.<ref>
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| {{Cite book | last = Chen | first = Wai-Kai | title = Graph Theory and its Engineering Applications | publisher = World Scientific | year = 1997 | pages = 29 | isbn = 978-981-02-1859-1}}</ref> A regular graph with vertices of degree <span class="texhtml" ><var >k</var ></span > is called a <span class="texhtml" ><var >k</var ></span >'''‑regular graph''' or regular graph of degree <span class="texhtml" ><var >k</var ></span >.
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| Regular graphs of degree at most 2 are easy to classify: A 0-regular graph consists of disconnected vertices, a 1-regular graph consists of disconnected edges, and a 2-regular graph consists of disconnected [[cycle (graph theory)|cycle]]s and infinite chains.
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| A 3-regular graph is known as a [[cubic graph]].
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| A [[strongly regular graph]] is a regular graph where every adjacent pair of vertices has the same number ''l'' of neighbors in common, and every non-adjacent pair of vertices has the same number ''n'' of neighbors in common. The smallest graphs that are regular but not strongly regular are the [[cycle graph]] and the [[circulant graph]] on 6 vertices.
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| The [[complete graph]] <math>K_m</math> is strongly regular for any <math>m</math>.
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| A theorem by [[Crispin St. J. A. Nash-Williams|Nash-Williams]] says that every <span class="texhtml" ><var >k</var ></span >‑regular graph on <span class="texhtml" >2<var >k</var > + 1</span > vertices has a [[Hamiltonian cycle]].
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| <gallery>
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| Image:0-regular_graph.svg|0-regular graph
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| Image:1-regular_graph.svg|1-regular graph
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| Image:2-regular_graph.svg|2-regular graph
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| Image:3-regular_graph.svg|3-regular graph
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| </gallery>
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| == Existence ==
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| It is well known that the necessary and sufficient conditions for a <math>k</math> regular graph of order <math>n</math> to exist are that <math> n \geq k+1 </math> and that <math> nk </math> is even. In such case it is easy to construct regular graphs by considering appropriate parameters for [[circulant graph]]s.
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| ==Algebraic properties==
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| Let ''A'' be the [[adjacency matrix]] of a graph. Then the graph is regular [[if and only if]] <math>\textbf{j}=(1, \dots ,1)</math> is an [[eigenvector]] of ''A''.<ref name="Cvetkovic">Cvetković, D. M.; Doob, M.; and Sachs, H. Spectra of Graphs: Theory and Applications, 3rd rev. enl. ed. New York: Wiley, 1998.</ref> Its eigenvalue will be the constant degree of the graph. Eigenvectors corresponding to other [[eigenvalue]]s are orthogonal to <math>\textbf{j}</math>, so for such eigenvectors <math>v=(v_1,\dots,v_n)</math>, we have <math>\sum_{i=1}^n v_i = 0</math>.
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| A regular graph of degree ''k'' is connected if and only if the eigenvalue ''k'' has multiplicity one.<ref name="Cvetkovic"/>
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| There is also a criterion for regular and connected graphs :
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| a graph is connected and regular if and only if the [[matrix of ones]] ''J'', with <math>J_{ij}=1</math>, is in the [[adjacency algebra]] of the graph (meaning it is a linear combination of powers of ''A'').<ref>{{citation
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| | last = Curtin | first = Brian
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| | doi = 10.1007/s10623-004-4857-4
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| | issue = 2-3
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| | journal = Designs, Codes and Cryptography
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| | mr = 2128333
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| | pages = 241–248
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| | title = Algebraic characterizations of graph regularity conditions
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| | volume = 34
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| | year = 2005}}.</ref>
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| Let G be a k-regular graph with diameter D and eigenvalues of adjacency matrix <math>k=\lambda_0 >\lambda_1\geq \dots\geq\lambda_{n-1}</math>. If G is not bipartite
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| <math>D\leq \frac{\log{(n-1)}}{\log(k/\lambda)}+1</math><ref name="personal.plattsburgh.edu">http://personal.plattsburgh.edu/quenelgt/pubpdf/diamest.pdf</ref>
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| where <math> \lambda=\max_{i>0}\{\mid \lambda_i \mid \}</math>.<ref name="personal.plattsburgh.edu"/>
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| == Generation ==
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| Regular graphs may be generated by the GenReg program.<ref>{{cite journal| last=Meringer | first=Markus | year=1999 | title=Fast generation of regular graphs and construction of cages | journal=[[Journal of Graph Theory]] | volume=30 | issue=2 | pages=137–146 | doi=10.1002/(SICI)1097-0118(199902)30:2<>1.0.CO;2-G | url=http://www.mathe2.uni-bayreuth.de/markus/pdf/pub/FastGenRegGraphJGT.pdf}}</ref>
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| == See also ==
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| * [[Random regular graph]]
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| * [[Strongly regular graph]]
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| * [[Moore graph]]
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| * [[Cage graph]]
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| == References ==
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| {{reflist}}
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| == External links ==
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| {{commons category|Regular graphs}}
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| * {{MathWorld|urlname=RegularGraph|title=Regular Graph}}
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| * {{MathWorld|urlname=StronglyRegularGraph|title=Strongly Regular Graph}}
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| * [http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html GenReg] software and data by Markus Meringer.
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| * {{Cite journal | last=Nash-Williams | first=Crispin |authorlink = Crispin St. J. A. Nash-Williams
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| | contribution=Valency Sequences which force graphs to have Hamiltonian Circuits
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| | title=University of Waterloo Research Report | publisher=University of Waterloo
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| | place=Waterloo, Ontario | year=1969 | postscript=<!--None--> }}
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| {{DEFAULTSORT:Regular Graph}}
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| [[Category:Graph families]]
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| [[Category:Regular graphs]]
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