Pseudo-Paley graphs and skew Hadamard difference sets from presemifields
Let (K, +, *) be an odd order presemifield with commutative multiplication. We show that the set of nonzero squares of (K, *) is a skew Hadamard difference set or a Paley type partial difference set in (K, +) according as q is congruent to 3 mod-ulo 4 or q is congruent to 1 modulo 4. Applying this result to the Coulter–Matthews presemifield and the Ding–Yuan variation of it, we recover a recent construction of skew Hadamard difference sets by Ding and Yuan . On the other hand, applying this result to the known presemifields with commutative multiplication and having order q congruent to 1 modulo 4, we construct several families of pseudo-Paley graphs. We compute the pranks of these pseudo-Paley graphs when q = 3 4 , 3 6 , 3 8 , 3 10 , 5 4 , and 7 4. The prank results indicate that these graphs seem to be new. Along the way, we also disprove a conjecture of René Peeters [17, p. 47] which says that the Paley graphs Dedicated to Dan Hughes on the occasion of his 80th birthday. of nonprime order are uniquely determined by their parameters and the minimality of their relevant pranks .