The minimum rank problem over finite fields

TL;DR

This paper characterizes all graphs with a minimum rank at most k over finite fields, linking the problem to projective geometry and deriving new results through this connection.

Contribution

It provides a comprehensive characterization of graphs with bounded minimum rank over finite fields and connects the problem to polarities in projective geometries.

Findings

Characterization of graphs with minimum rank ≤ k over finite fields

Establishment of a connection between minimum rank and projective geometry polarities

Derivation of new results using known geometric properties

Abstract

The structure of all graphs having minimum rank at most k over a finite field with q elements is characterized for any possible k and q. A strong connection between this characterization and polarities of projective geometries is explained. Using this connection, a few results in the minimum rank problem are derived by applying some known results from projective geometry.