A strongly regular graph with parameters has graph eigenvalues , , and , where
(1)
| |||
(2)
|
where
(3)
|
(Godsil and Royle 2001, pp. 221-222). In the case of and distinct, call their multiplicities in the graph spectrum and . Then a graph with is called a conference graph. All Paley graphs are conference graphs.
A strongly regular graph is either a conference graph, has and integers and a perfect square (correcting a typo in Godsil and Royle 2001, p. 222), or both of the above (Godsil and Royle 2001, p. 222). Paley graphs with a square number (including the (2,1)-generalized quadrangle, which is isomorphic to the 9-Paley graph) satisfy both conditions.
In the special case that is a strongly regular graph with vertices where is prime, is a conference graph (Godsil and Royle 2001, p. 222).
The following table summarizes some conference graphs.
graph | characteristic polynomial | ||
5 | 5-cycle graph | ||
9 | -generalized quadrangle | ||
13 | 13-Paley | ||
17 | 17-Paley | ||
25 | 25-Paley | ||
25 | 25-Paley | ||
25 | 25-Paulus graph 1-14 | ||
29 | 29-Paley | ||
37 | 37-Paley | ||
41 | 41-Paley | ||
49 | 49-Paley | ||
53 | 53-Paley | ||
61 | 61-Paley | ||
73 | 73-Paley | ||
81 | 81-Paley | ||
89 | 89-Paley | ||
97 | 97-Paley | ||
101 | 101-Paley | ||
109 | 109-Paley | ||
113 | 113-Paley | ||
121 | 121-Paley | ||
125 | 125-Paley | ||
137 | 137-Paley | ||
149 | 149-Paley | ||
157 | 157-Paley | ||
169 | 169-Paley |