A strongly regular graph is a 
-regular simple graph 
on 
nodes for which there exist positive integers 
, 
, and 
such that every vertex has 
neighbors (i.e., the graph is a regular
graph), every adjacent pair of vertices has 
common neighbors, and every nonadjacent pair has 
common neighbors (West 2000, pp. 464-465).
A graph that is not strongly regular is said to be weakly
regular.
The parameter 
here is unrelated to the 
used for the mutual-visibility
number, the 
of the matching polynomial, and the 
used for circuit rank.
A distance-regular graph with graph diameter 
is a strongly regular graph (Biggs 1993, p. 159). Strongly regular graphs are
therefore distance-regular. Connected strongly
regular graphs are conformally rigid (Steinerberger
and Thomas 2024).
The complete graph 
is strongly regular for all 
. The status of the trivial singleton
graph 
is unclear. Opinions differ on if 
is a strongly regular graph, though since it has no well-defined
parameter, it is preferable to consider it not to be strongly regular (A. E. Brouwer,
pers. comm., Feb. 6, 2013).
The graph complement of a non-empty non-complete strongly regular graph with parameters 
is another strongly regular graph with parameters
.
A number of strongly regular graphs are implemented in the Wolfram Language as GraphData["StronglyRegular"].
The 
strongly regular graph and its graph complement,
the 
strongly regular graph, both have the Mathieu group
M22, of order 
, as their automorphism
group (DistanceRegular.org). The 
strongly regular graph has the Mathieu
group M23, of order 
, as its automorphism
group (DistanceRegular.org).
The numbers of strongly regular graphs on 
, 2, ... nodes are 1, 1, 2, 4, 3, 6, 2, 6, 5, ... (OEIS
A076435), the first few of which are illustrated
above. The smallest regular graphs that are not
strongly regular are the cycle graph 
and circulant graph 
.
Similarly, the numbers of connected strongly regular graphs on 
, 2, ... nodes are 1, 0, 1, 2, 2, 3, 1, 3, 3, ... (OEIS
A088741).
Brouwer (2013) has conjectured that all connected strongly regular graphs (where 
is assumed to not be strongly regular) are Hamiltonian
with the exception of the Petersen graph.
Other than the trivial singleton graph 
and the complete
bipartite graphs 
, there are exactly seven known connected triangle-free
strongly regular graphs, as summarized in the following table (Godsil 1995) and six
of which are illustrated above. Determining the existence or absence of any others
remains an open problem.
 | graph |
| 5 | 5-cycle graph |
| 10 | Petersen graph |
| 16 | Clebsch graph |
| 50 | Hoffman-Singleton graph |
| 56 | Gewirtz graph |
| 77 | M22 graph |
| 100 | Higman-Sims graph |
Examples of connected non-complete strongly regular graphs are given in the following table.
graph
graph
Strongly regular graphs with 
correspond to symmetric balanced incomplete block
designs (West 2000, p. 465). Strongly regular graphs also arise as point
graphs of partial geometries. In this context,
a graph with the corresponding parameter set is a pseudogeometric
graph; when the incidence structure exists
and its lines give the Delsarte
cliques, the graph is a geometric graph. For
example, the McLaughlin graph has the parameters
of a point graph of a partial
geometry 
,
but the corresponding McLaughlin geometry
does not exist (Östergård and Soicher 2016).
Graph." 
(40 graphs, 20 pairs)." 
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."