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| [[Image:Paley13.svg|thumb|240px|The [[Paley graph]] of order 13, a strongly regular graph with parameters srg(13,6,2,3).]]
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| {{Graph families defined by their automorphisms}}
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| In [[graph theory]], a '''strongly regular graph''' is defined as follows. Let ''G'' = (''V'',''E'') be a [[regular graph]] with ''v'' vertices and degree ''k''. ''G'' is said to be '''strongly regular''' if there are also [[integer]]s λ and μ such that:
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| * Every two [[adjacent vertices]] have λ common neighbours.
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| * Every two non-adjacent vertices have μ common neighbours.
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| A graph of this kind is sometimes said to be an srg(''v'', ''k'', λ, μ).
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| Some authors exclude graphs which satisfy the definition trivially, namely those graphs which are the disjoint union of one or more equal-sized [[complete graph]]s,<ref>[http://homepages.cwi.nl/~aeb/math/ipm.pdf Brouwer, Andries E; Haemers, Willem H. ''Spectra of Graphs''. p. 101]</ref><ref>Godsil, Chris; Royle, Gordon. ''Algebraic Graph Theory''. Springer-Verlag New York, 2001, p. 218.</ref> and their [[complement graph|complements]], the [[Turán graph]]s.
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| The [[complement graph|complement]] of an srg(''v'', ''k'', λ, μ) is also strongly regular. It is an srg(''v'', ''v−k''−1, ''v''−2−2''k''+μ, ''v''−2''k''+λ).
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| A strongly regular graph is a [[distance-regular graph]] with diameter 2, but only if μ is non-zero.
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| ==Properties==
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| ===Relationship between Parameters===
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| The four parameters in an srg(''v'', ''k'', λ, μ) are not independent and must obey the following relation:
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| :<math>(v-k-1)\mu = k(k-\lambda-1)</math>
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| The above relation can be derived very easily through a counting argument as follows:
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| * Imagine the nodes of the graph to lie in three levels. Pick any node as the root node, in Level 0. Then its ''k'' neighbor nodes lie in Level 1, and all other nodes lie in Level 2.
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| * Nodes in Level 1 are directly connected to the root, hence they must have λ other neighbors in common with the root, and these common neighbors must also be in Level 1. Since each node has degree ''k'', there are <math>k-\lambda-1</math> edges remaining for each Level 1 node to connect to nodes in Level 2.
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| * Nodes in Level 2 are not directly connected to the root, hence they must have μ common neighbors with the root, and these common neighbors must all be in Level 1. Therefore μ nodes in Level 1 are connected to each node in Level 2 and each of the ''k'' nodes in Level 1 is connected to <math>k-\lambda-1</math> nodes in Level 2. Therefore the number of nodes in Level 2 is
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| ::<math>\frac{k(k-\lambda-1)}{\mu}.</math>
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| The total number of nodes across all three levels is therefore
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| :<math>v = 1 + k + \left (\frac{k(k-\lambda-1)}{\mu} \right ),</math>
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| and by rearranging we get:
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| :<math>(v-k-1)\mu = k(k-\lambda-1).</math>
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| ===Adjacency Matrix===
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| Let ''I'' denote the identity matrix (of order ''v'') and let ''J'' denote the matrix whose entries all equal 1. The [[adjacency matrix]] ''A'' of a strongly regular graph satisfies these properties :
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| :<math>AJ = kJ</math>
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| This is a trivial restatement of the vertex degree requirement.
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| :<math>{A}^{2} + (\mu-\lambda){A} + (\mu-k){I} = \mu {J}</math>
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| The first term gives the number of 2-step paths from each vertex to all vertices. For the vertex pairs directly connected by an edge, the equation reduces to the number of such 2-step paths being equal to λ. For the vertex pairs not directly connected by an edge, the equation reduces to the number of such 2-step paths being equal to μ. For the trivial self-pairs, the equation reduces to the degree being equal to ''k''.
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| ===Eigenvalues===
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| *The graph has exactly three [[eigenvalue]]s:
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| **''k'' whose [[Multiplicity (mathematics)|multiplicity]] is 1
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| ** <math>\frac{1}{2}\left[(\lambda-\mu)+\sqrt{(\lambda-\mu)^2 + 4(k-\mu)}\right]</math> whose multiplicity is <math>\frac{1}{2} \left[(v-1)-\frac{2k+(v-1)(\lambda-\mu)}{\sqrt{(\lambda-\mu)^2 + 4(k-\mu)}}\right]</math>
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| ** <math>\frac{1}{2}\left[(\lambda-\mu)-\sqrt{(\lambda-\mu)^2 + 4(k-\mu)}\right]</math> whose multiplicity is <math>\frac{1}{2} \left[(v-1)+\frac{2k+(v-1)(\lambda-\mu)}{\sqrt{(\lambda-\mu)^2 + 4(k-\mu)}}\right]</math>
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| * Strongly regular graphs for which <math>2k+(v-1)(\lambda-\mu) = 0</math> are called [[conference graph]]s because of their connection with symmetric [[conference matrix|conference matrices]]. Their parameters reduce to
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| ::<math>\text{srg}\left(v, \tfrac{1}{2}(v-1), \tfrac{1}{4}(v-5), \tfrac{1}{4}(v-1)\right).</math>
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| * Strongly regular graphs for which <math>2k+(v-1)(\lambda-\mu) \ne 0</math> have integer eigenvalues with unequal multiplicities.
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| ==Examples==
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| * The [[Cycle graph|cycle]] of length 5 is an srg(5,2,0,1).
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| * The [[Petersen graph]] is an srg(10,3,0,1).
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| * The [[Clebsch graph]] is an srg(16,5,0,2).
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| * The [[Shrikhande graph]] is an srg(16,6,2,2) which is not a [[distance-transitive graph]].
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| * The [[Line graph]] of [[generalized quadrangle]] GQ(2,4) are srg(27,10,1,5).
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| * The [[Schläfli graph]] is an srg(27,16,10,8).<ref>{{MathWorld | urlname=SchlaefliGraph | title=Schläfli graph}}</ref>
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| * The [[Chang graphs]] are srg(28,12,6,4).
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| * The [[Hoffman–Singleton graph]] is an srg(50,7,0,1).
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| * The [[Sims-Gewirtz graph]] is an (56,10,0,2).
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| * The [[M22 graph]] is an srg(77,16,0,4).
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| * The [[Brouwer–Haemers graph]] is an srg(81,20,1,6).
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| * The [[Higman–Sims graph]] is an srg(100,22,0,6).
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| * The [[Local McLaughlin graph]] is an srg(162,56,10,24).
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| * The [[Paley graph]] of order ''q'' is an srg(''q'', (''q'' − 1)/2, (''q'' − 5)/4, (''q'' − 1)/4).
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| * The ''n'' × ''n'' square [[rook's graph]] is an srg(''n''<sup>2</sup>, 2''n'' − 2, ''n'' − 2, 2).
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| A strongly regular graph is called '''primitive''' if both the graph and its complement are connected. All the above graphs are primitive, as otherwise μ=0 or μ=k.
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| ===Moore graphs===
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| The strongly regular graphs with λ=0 are [[triangle-free graph|triangle free]]. Apart from the complete graphs on less than 3 vertices and all complete bipartite graphs the seven listed above are the only known ones. Strongly regular graphs with λ=0 and μ=1 are [[Moore graph]]s with girth 5. Again the three graphs given above, with parameters (5,2,0,1), (10,3,0,1) and (50,7,0,1), are the only known ones. The only other possible set of parameters yielding a Moore graph is (3250,57,0,1); it is unknown if such a graph exists, and if so, whether or not it is unique.
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| ==See also==
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| * [[Seidel adjacency matrix]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * A.E. Brouwer, A.M. Cohen, and A. Neumaier (1989), ''Distance Regular Graphs''. Berlin, New York: Springer-Verlag. ISBN 3-540-50619-5, ISBN 0-387-50619-5
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| * [[Chris Godsil]] and Gordon Royle (2004), ''Algebraic Graph Theory''. New York: Springer-Verlag. ISBN 0-387-95241-1
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| ==External links==
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| * [[Eric W. Weisstein]], [http://mathworld.wolfram.com/StronglyRegularGraph.html Mathworld article with numerous examples.]
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| * [[Gordon Royle]], [http://people.csse.uwa.edu.au/gordon/remote/srgs/ List of larger graphs and families.]
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| * [[Andries E. Brouwer]], [http://www.win.tue.nl/~aeb/graphs/srg/srgtab.html Parameters of Strongly Regular Graphs.]
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| * [[Brendan McKay]], [http://cs.anu.edu.au/~bdm/data/graphs.html Some collections of graphs.]
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| * [[Ted Spence]], [http://www.maths.gla.ac.uk/~es/srgraphs.php Strongly regular graphs on at most 64 vertices.]
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| {{DEFAULTSORT:Strongly Regular Graph}}
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| [[Category:Graph families]]
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| [[Category:Algebraic graph theory]]
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| [[Category:Regular graphs]]
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