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Diameter preserving surjections in the geometry of matrices

Wen-ling Huang, Hans Havlicek

Abstract

We consider a class of graphs subject to certain restrictions, including the finiteness of diameters. Any surjective mapping $φ:Γ\toΓ'$ between graphs from this class is shown to be an isomorphism provided that the following holds: Any two points of $Γ$ are at a distance equal to the diameter of $Γ$ if, and only if, their images are at a distance equal to the diameter of $Γ'$. This result is then applied to the graphs arising from the adjacency relations of spaces of rectangular matrices, spaces of Hermitian matrices, and Grassmann spaces (projective spaces of rectangular matrices).

Diameter preserving surjections in the geometry of matrices

Abstract

We consider a class of graphs subject to certain restrictions, including the finiteness of diameters. Any surjective mapping between graphs from this class is shown to be an isomorphism provided that the following holds: Any two points of are at a distance equal to the diameter of if, and only if, their images are at a distance equal to the diameter of . This result is then applied to the graphs arising from the adjacency relations of spaces of rectangular matrices, spaces of Hermitian matrices, and Grassmann spaces (projective spaces of rectangular matrices).

Paper Structure

This paper contains 6 sections, 9 theorems, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The main result
  3. Applications
  4. Geometry of rectangular matrices
  5. Geometry of Hermitian and symmetric matrices
  6. Projective geometry of rectangular matrices--- the Grassmann space

Key Result

Lemma 2.1

Given a graph $\Gamma$ which satisfies conditions (A1)--(A4) let $n:=\mathop{\rm diam}\nolimits \Gamma$. Suppose that $a,b\in{\mathcal{P}}$ are distinct points with the following property: Then $a$ and $b$ are adjacent.

Figures (2)

  • Figure 1: A counterexample for (A4) and Lemma \ref{['lem:A1-4']}
  • Figure 2: A counterexample for (A5)

Theorems & Definitions (18)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 8 more