Good distance graphs and the geometry of matrices

Abstract

Denote by G = ( V , ∼ ) a graph which V is the vertex set and ∼ is an adjacency relation on a subset of V × V . In this paper, the good distance graph is defined. Let ( V , ∼ ) and ( V ′ , ∼ ′ ) be two good distance graphs, and φ : V → V ′ be a map. The following theorem is proved: φ is a graph isomorphism ⇔ φ is a bounded distance preserving surjective map in both directions ⇔ φ is a distance k preserving surjective map in both directions (where k < diam ( G ) / 2 is a positive integer), etc. Let D be a division ring with an involution ¯ such that both | F ∩ Z D | ⩾ 3 and D is not a field of characteristic 2 with D = F , where F = { a ∈ D : a = a ¯ } and Z D is the center of D . Let H n ( n ⩾ 2 ) be the set of n × n Hermitian matrices over D . It is proved that ( H n , ∼ ) is a good distance graph, where A ∼ B ⇔ rank ( A - B ) = 1 for all A , B ∈ H n .