A strongly regular graph with parameters has graph eigenvalues , , and , where
(1)
 
(2)

where
(3)

(Godsil and Royle 2001, pp. 221222). 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 9Paley 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  5cycle graph  
9  generalized quadrangle  
13  13Paley  
17  17Paley  
25  25Paley  
25  25Paley  
25  25Paulus graph 114  
29  29Paley  
37  37Paley  
41  41Paley  
49  49Paley  
53  53Paley  
61  61Paley  
73  73Paley  
81  81Paley  
89  89Paley  
97  97Paley  
101  101Paley  
109  109Paley  
113  113Paley  
121  121Paley  
125  125Paley  
137  137Paley  
149  149Paley  
157  157Paley  
169  169Paley 