2025/08/10 by Jing Zeng, Lihua You, Zeng, Jing +4
Pharmacology, Toxicology and Pharmaceutics · Business, Management and Accounting · #Pharmacy and Medical Practices #Franchising Strategies and Performance
paper · pdf · doi:10.48550/arxiv.2508.07332
The determinant of a tournament T, denoted by det(T), is defined as the determinant of the skew-adjacency matrix of T. It is well-known that det(T) is equal to 0 if n is odd, and det(T) is the square of an odd integer if n is even. For a positive odd integer k, let Dk be the set of tournaments whose all subtournaments have determinant at most k2. Former studies showed that for k ∈ \1,3,5\, a tournament T ∈ Dk \backslash Dk-2 (T ∈ D1 when k=1) if and only if T is switching equivalent to a transitive blowup of Lk+1, where Lk+1 is a tournament of order k+1 with a specific structure. For k ≥ 7, no characterization results are known. A natural problem is to characterize tournaments in Dk \backslash Dk-2 that can be switching equivalent to a transitive blowup of Lk+1 for k ≥ 7. To address this problem and to further explore the structural properties of tournaments in Dk, we introduce CR tournaments, strong CR tournaments, basic tournaments and Z-matrices, and investigate their properties. We use these properties to characterize those tournaments T ∈ Dk \backslash Dk-2 where T contains a subtournament switching isomorphic to a basic strong CR tournament in Dk \backslash Dk-2. This result implies former characterizations of D3\backslash D1 and D5 \backslash D3. Using Z-matrices, we also show that for even n, Ln is a basic strong CR tournament, and thus solve the open problem posed in [Discrete Math. 349 (2) (2026) 114766].