Discrete Mathematics With Graph Theory 7.7 21 Answers
Strongly Regular Graph
A 
-regular simple graph 
on 
nodes is strongly 
-regular if 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 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 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 |
 | square graph |
 | cycle graph  |
 | utility graph |
 | octahedral graph |
 | complete bipartite graph  |
 | 16-cell graph |
 | generalized quadrangle GQ(2,1) |
 | complete tripartite graph  |
 | Petersen graph |
 | complete bipartite graph  |
 | 5-triangular graph |
 | 5-cocktail party graph |
 | (6,6)-complete bipartite graph  |
 | (4,4,4)-complete tripartite graph  |
 | (3,3,3,3)-complete 4-partite graph  |
 | 6-cocktail party graph |
 | 13-Paley graph |
 | complete bipartite graph  |
 | 7-cocktail party graph |
 | generalized quadrangle GQ(2,2) |
 | 6-triangular graph |
 | complete tripartite graph  |
 | complete 5-partite graph  |
 | Clebsch graph |
 | (4,4)-rook graph, Shrikhande graph |
 | complete bipartite graph  |
 | complement of (4,4)-rook graph |
 | 5-halved cube graph |
 | complete 4-partite graph  |
 | 8-cocktail party graph |
 | 17-Paley graph |
 | complete bipartite graph  |
 | complete tripartite graph  |
 | 9-cocktail party graph |
 | complete bipartite graph  |
 | 10-cocktail party graph |
 | (7,2)-Kneser graph |
 | 7-triangular graph |
 | complete bipartite graph  |
 | 11-cocktail party graph |
 | complete bipartite graph  |
 | 12-cocktail party graph |
 | (5,5)-rook graph |
 | 25-Paley graph, 25-Paulus graphs |
 | 26-Paulus graphs |
 | complete bipartite graph  |
 | 13-cocktail party graph |
 | generalized quadrangle GQ(2,4) |
 | Schläfli graph |
 | 8-triangular graph, Chang graphs |
 | complete bipartite graph  |
 | (8,2)-Kneser graph |
 | 14-cocktail party graph |
 | (29,14,6,7)-strongly regular graphs, 29-Paley graph |
 | complete bipartite graph  |
 | 15-cocktail party graph |
 | complete bipartite graph  |
 | 16-cocktail party graph |
 | complete bipartite graph  |
 | 17-cocktail party graph |
 | (6,6)-rook graph |
 |  graph |
 | 9-triangular graph |
 | complete bipartite graph  |
 | (9,2)-Kneser graph |
 | 18-cocktail party graph |
 | 37-Paley graph |
 | complete bipartite graph  |
 | 19-cocktail party graph |
 | complete bipartite graph  |
 | 20-cocktail party graph |
 | 41-Paley graph |
 | 10-triangular graph |
 | (10,2)-Kneser graph |
 | (7,7)-rook graph |
 | 49-Paley graph |
 | Hoffman-Singleton graph |
 | Hoffman-Singleton graph complement |
 | 53-Paley graph |
 | 11-triangular graph |
 | (11,2)-Kneser graph |
 | Gewirtz graph |
 | 61-Paley graph |
 | (63,32,16,16)-strongly regular graph |
 | (8,8)-rook graph |
 | 64-cyclotomic graph |
 | 12-triangular graph |
 | (12,2)-Kneser graph |
 | 73-Paley graph |
 | M22 graph |
 | 13-triangular graph |
 | (13,2)-Kneser graph |
 | (9,9)-rook graph |
 | Brouwer-Haemers graph |
 | 81-Paley graph |
 | 89-Paley graph |
 | 14-triangular graph |
 | (14,2)-Kneser graph |
 | 97-Paley graph |
 | (10,10)-rook graph |
 | Higman-Sims graph |
 | Hall-Janko graph |
 | 101-Paley graph |
 | 15-triangular graph |
 | (15,2)-Kneser graph |
 | 109-Paley graph |
 | generalized quadrangle GQ(3,9) |
 | 113-Paley graph |
 | 16-triangular graph |
 | (120,56,28,24)-strongly regular graph |
 | (120,63,30,36)-strongly regular graph |
 | 121-Paley graph |
 | 125-Paley graph |
 | 17-triangular graph |
 | 137-Paley graph |
 | 149-Paley graph |
 | 18-triangular graph |
 | 157-Paley graph |
 | local McLaughlin graph |
 | 169-Paley graph |
 | 19-triangular graph |
 | 20-triangular graph |
 | Berlekamp-van Lint-Seidel graph |
 | Delsarte graph |
 | (253,112,36,60)-strongly regular graph |
 | McLaughlin graph |
 |  graph |
 | Games graph |
Strongly regular graphs with 
correspond to symmetric balanced incomplete block designs (West 2000, p. 465).
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Discrete Mathematics With Graph Theory 7.7 21 Answers
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