It is the graph associated with a symmetric conference matrix, and consequently its order v must be 1 (modulo 4) and a sum of two squares.
Conference graphs are known to exist for all small values of v allowed by the restrictions, e.g., v = 5, 9, 13, 17, 25, 29, and (the Paley graphs) for all prime powers congruent to 1 (modulo 4).
However, there are many values of v that are allowed, for which the existence of a conference graph is unknown.
The eigenvalues of a conference graph need not be integers, unlike those of other strongly regular graphs.
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