Some applications of linear algebraic methods in combinatorics

TL;DR

This paper explores the use of linear algebraic methods to generate Latin squares and 4-cycle systems by identifying basis vectors through kernel analysis of associated matrices.

Contribution

It introduces a novel approach to construct Latin squares and 4-cycle systems using linear algebraic techniques involving kernel bases of specific matrices.

Findings

All Latin squares can be generated from a single square using small trades.

The method applies to 4-cycle systems similarly.

The approach links combinatorial structures to graph-based matrix analysis.

Abstract

In this note, we intend to produce all latin squares from one of them using suitable move which is defined by small trades and do the similar work on 4-cycle systems. These problems, reformulate as finding basis for the kernel of special matrices, representef to some graphs.