On cyclotomic nearly-doubly-regular tournaments

Abstract: Nearly-doubly-regular tournaments have played significant roles in extremal graph theory. In this note, we construct new cyclotomic nearly-doubly-regular tournaments and determine their spectrum by establishing a new connection between cyclotomic nearly-doubly-regular tournaments and almost difference sets from combinatorial design theory. Furthermore, under the celebrated Hardy-Littlewood conjecture F in analytic number theory, our results confirm the conjecture due to Sergey Savchenko (J. Graph Theory {\bf 83} (2016), 44--77) on the existence of infinitely many nearly-doubly-regular tournaments with the canonical spectrum.