Editing Strongly regular graph (section)

==Examples==
* The [[Cycle graph|cycle]] of length 5 is an srg(5, 2, 0, 1).
* The [[Petersen graph]] is an srg(10, 3, 0, 1).
* The [[Clebsch graph]] is an srg(16, 5, 0, 2).
* The [[Shrikhande graph]] is an srg(16, 6, 2, 2) which is not a [[distance-transitive graph]].
* The ''n''&nbsp;×&nbsp;''n'' square [[rook's graph]], i.e., the line graph of a balanced complete [[bipartite graph]] ''K''<sub>''n'',''n''</sub>, is an srg(''n''<sup>2</sup>, 2''n''&nbsp;−&nbsp;2, ''n''&nbsp;−&nbsp;2, 2).  The parameters for {{nowrap|''n'' {{=}} 4}} coincide with those of the Shrikhande graph, but the two graphs are not isomorphic. (The vertex neighborhood for the Shrikhande graph is a hexagon, while that for the rook graph is two triangles.)
* The [[line graph]] of a complete graph ''K<sub>n</sub>'' is an <math display="inline">\operatorname{srg}\left(\binom{n}{2}, 2(n - 2), n - 2, 4\right)</math>.
* The three [[Chang graphs]] are srg(28, 12, 6, 4), the same as the line graph of ''K''<sub>8</sub>, but these four graphs are not isomorphic.
* Every [[generalized quadrangle]] of order (s, t) gives an srg((s + 1)(st + 1), s(t + 1), s − 1, t + 1) as its [[line graph]]. For example, GQ(2, 4) gives srg(27, 10, 1, 5) as its line graph.
* The [[Schläfli graph]] is an srg(27, 16, 10, 8) and is the complement of the aforementioned line graph on GQ(2, 4).<ref>{{MathWorld | urlname=SchlaefliGraph | title=Schläfli graph|mode=cs2}}</ref>
* The [[Hoffman–Singleton graph]] is an srg(50, 7, 0, 1).
* The [[Gewirtz graph]] is an srg(56, 10, 0, 2).
* The [[M22 graph]] aka the [[Mesner graph]] is an srg(77, 16, 0, 4).
* The [[Brouwer–Haemers graph]] is an srg(81, 20, 1, 6).
* The [[Higman–Sims graph]] is an srg(100, 22, 0, 6).
* The [[Local McLaughlin graph]] is an srg(162, 56, 10, 24).  
* The [[Cameron graph]] is an srg(231, 30, 9, 3).
* The [[Berlekamp–van Lint–Seidel graph]] is an srg(243, 22, 1, 2).
* The [[McLaughlin graph]] is an srg(275, 112, 30, 56).
* The [[Paley graph]] of order ''q'' is an srg(''q'', (''q''&nbsp;−&nbsp;1)/2, (''q''&nbsp;−&nbsp;5)/4, (''q''&nbsp;−&nbsp;1)/4).  The smallest Paley graph, with {{nowrap|''q'' {{=}} 5}}, is the 5-cycle (above).
* [[Self-complementary graph|Self-complementary]] [[symmetric graph|arc-transitive]] graphs are strongly regular.

A strongly regular graph is called '''primitive''' if both the graph and its complement are connected. All the above graphs are primitive, as otherwise {{nowrap|μ {{=}} 0}} or {{nowrap|λ {{=}} ''k''}}.

[[Conway's 99-graph problem]] asks for the construction of an srg(99, 14, 1, 2). It is unknown whether a graph with these parameters exists, and [[John Horton Conway]] offered a $1000 prize for the solution to this problem.<ref>{{citation
 | last = Conway | first = John H. | author-link = John Horton Conway
 | accessdate = 2019-02-12
 | publisher = Online Encyclopedia of Integer Sequences
 | title = Five $1,000 Problems (Update 2017)
 | url = https://oeis.org/A248380/a248380.pdf
}}</ref>

===Triangle-free graphs===
The strongly regular graphs with λ&nbsp;=&nbsp;0 are [[triangle-free graph|triangle free]]. Apart from the complete graphs on fewer than 3 vertices and all complete bipartite graphs, the seven listed earlier (pentagon, Petersen, Clebsch, Hoffman-Singleton, Gewirtz, Mesner-M22, and Higman-Sims) are the only known ones.

===Geodetic graphs===
Every strongly regular graph with <math>\mu = 1</math> is a [[geodetic graph]], a graph in which every two vertices have a unique [[Shortest path problem|unweighted shortest path]].<ref name=bb>{{citation
 | last1 = Blokhuis | first1 = A.
 | last2 = Brouwer | first2 = A. E. | authorlink = Andries Brouwer
 | doi = 10.1007/BF00191941
 | issue = 1–3
 | journal = [[Geometriae Dedicata]]
 | mr = 925851
 | pages = 527–533
 | title = Geodetic graphs of diameter two
 | volume = 25
 | year = 1988
| s2cid = 189890651
 }}</ref> The only known strongly regular graphs with <math>\mu = 1</math> are those where <math>\lambda</math> is 0, therefore triangle-free as well. These are called the Moore graphs and are [[#The Hoffman–Singleton theorem|explored below in more detail]]. Other combinations of parameters such as (400, 21, 2, 1) have not yet been ruled out. Despite ongoing research on the properties that a strongly regular graph with <math>\mu=1</math> would have,<ref>{{citation
 | last1 = Deutsch | first1 = J.
 | last2 = Fisher | first2 = P. H.
 | doi = 10.1006/eujc.2000.0472
 | issue = 3
 | journal = [[European Journal of Combinatorics]]
 | mr = 1822718
 | pages = 303–306
 | title = On strongly regular graphs with <math>\mu=1</math>
 | volume = 22
 | year = 2001| doi-access = free
}}</ref><ref>{{citation
 | last1 = Belousov | first1 = I. N.
 | last2 = Makhnev | first2 = A. A.
 | issue = 2
 | journal = [[Doklady Akademii Nauk]]
 | mr = 2455371
 | pages = 151–155
 | title = On strongly regular graphs with <math>\mu=1</math> and their automorphisms
 | volume = 410
 | year = 2006
}}</ref> it is not known whether any more exist or even whether their number is finite.<ref name=bb/> Only the elementary result is known, that <math>\lambda</math> cannot be 1 for such a graph.