Diameter preserving surjections in the geometry of matrices

TL;DR

This paper proves that certain distance-preserving surjective maps between graphs, including those from matrix spaces, are actually isomorphisms, under specific diameter-related conditions.

Contribution

It establishes that diameter-preserving surjections between restricted graphs are isomorphisms, with applications to matrix and Grassmannian graphs.

Findings

Surjective maps preserving diameter-related distances are isomorphisms.

Results apply to graphs from matrix spaces and Grassmann spaces.

Provides a characterization of structure-preserving maps in these geometries.

Abstract

We consider a class of graphs subject to certain restrictions, including the finiteness of diameters. Any surjective mapping $\phi:\Gamma\to\Gamma'$ between graphs from this class is shown to be an isomorphism provided that the following holds: Any two points of $\Gamma$ are at a distance equal to the diameter of $\Gamma$ if, and only if, their images are at a distance equal to the diameter of $\Gamma'$. 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).