An algebraic characterization of finite symmetric tournaments

Abstract

A tournament T is called symmetric if its automorphism groupis transitive on the points and arcs of 1 . The main result of this paper is that if T is a finite symmetric tournament then T is isomorphic to one of the quadratic residue tournaments formed on the points of a finite field GF (p n ) ,

Paley and the Paley Graphs

“…where p = 4k + 1 and n is the residue of n mod (p) closest to 0, and also in [83] some efficient modern algorithms for solving (4). In 1907 Jacobsthal [45] published explicit formulae for integers a and b satisfying (4), based on work in his thesis [44].…”

-Transitive Digraphs Preserving a Cartesian Decomposition

“…As |V * | = q 2 − 1, we obtain that ∆ contains 2(q + 1) elements. For (a, b) ∈ V * , we denote by [ [1,0]). Therefore X q is a directed graph of in-and out-valency q.…”

“…, 0] is the unique vertex of X q not adjacent to [1,0]. Now vertex transitivity shows that, for each vertex v, there exists a unique vertex which is not adjacent to v (namely v z ).…”

“…Proof. Berggren [1] shows that if T is a finite symmetric tournament, then T is isomorphic to T q for some q ≡ 3 mod 4. In particular, we may assume that T = T q .…”

“…In particular, we may assume that T = T q . Moreover, [1,Theorem A] gives that the automorphism group Aut(T q ) of T q is the group of all affine permutations of F q of the form τ σ,x 2 ,c : a → x 2 a σ + c, where c ∈ F q , x ∈ F q \{0}, and σ ∈ Gal(F q ). Using this description of Aut(T q ), it is easy to see that if H acts transitively on the arcs of T q , then A = {τ id,x 2 ,c : x, c ∈ F q , x 0} is a subgroup of H (where id denotes the identity Galois automorphism of F q ).…”