2014/12/05 by Oleksiy Dovgoshey, Dovgoshey, O., Evgeniy Petrov +3 · 1 citation
Mathematics · #37E25 #54E35 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.1412.1979
openalex publication_date 2014/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (X,d) be a finite ultrametric space. In 1961 E.C. Gomory and T.C. Hu proved the inequality |Sp(X)|\leqslant |X| where Sp(X)=\d(x,y)\colon x,y ∈ X\. Using weighted Hamiltonian cycles and weighted Hamiltonian paths we give new necessary and sufficient conditions under which the Gomory-Hu inequality becomes an equality. We find the number of non-isometric (X,d) satisfying the equality |Sp(X)|=|X| for given Sp(X). Moreover it is shown that every finite semimetric space Z is an image under a composition of mappings f\colon X→ Y and g\colon Y→ Z such that X and Y are finite ultrametric space, X satisfies the above equality, f is an ε-isometry with an arbitrary ε>0, and g is a ball-preserving map.