Good distance graphs and the geometry of matrices

✍ Scribed by Li-Ping Huang

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 234 KB
• Volume: 433
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2010.02.013

✦ Synopsis

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 = ā} 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 .