On the independence number of regular graphs of matrix rings

TL;DR

This paper investigates a graph formed by non-singular matrices over a finite field, establishing an upper bound on its independence number and deriving a lower bound on its chromatic number, advancing understanding of matrix ring graph properties.

Contribution

It provides a new upper bound for the independence number of a matrix ring graph and improves the lower bound for its chromatic number, enhancing prior results.

Findings

Upper bound for the independence number established

Lower bound for the chromatic number improved

Results advance understanding of matrix ring graph properties

Abstract

Consider a graph on the non-singular matrices over a finite field, in which two distinct non-singular matrices are joined by an edge whenever their sum is singular. We prove an upper bound for the independence number of this graph. As a consequence, we obtain a lower bound for its chromatic number that significantly improves a previous result of Tomon.