On the number of 8-cycles for two particular regular tournaments of order N with diametrically opposite local properties

For a regular tournament T of order $n,$ denote by c₈(T) the number of cycles of length $8$ in $T.$ Let DRₙ be a doubly-regular tournament of order n≡ 34 (so, the out-sets and in-sets of its vertices are also regular and hence, contain the maximum possible number of cyclic triples) and RLTₙ be the unique regular locally transitive tournament of (odd) order n (so, the out-sets and in-sets of its vertices are transitive and hence, contain no cyclic triples, at all). Some arguments based on the spectral properties of tournaments allow us to suggest that c₈(T) ≤ c₈(RLTₙ), where n is sufficiently large. This restriction on n is essential because our computer processing of B. McKay's file of tournaments implies that for $n=9,11,13,$ the maximum of c₈(T) is attained at tournaments with regular structure of the out and in-sets of their vertices. In the present paper, we show that c₈(DRₙ) does not depend on a particular choice of DRₙ and determine expressions for c₈(DRₙ) and c₈(RLTₙ). They are both polynomials of degree $8$ in $n.$ Comparing c₈(DRₙ) with c₈(RLTₙ) yields the inequality c₈(DRₙ)>c₈(RLTₙ) for 11≤ n≤ 35, while c₈(RLTₙ) > c₈(DRₙ) for n≥ 39. This allows us to treat the value $n=39$ as the point of phase transition in the local properties of maximizers and minimizers of c₈(T) in the class of regular tournaments of order $n.$