# Strongly regular decompositions and symmetric association schemes of a power of two

## TL;DR

This paper constructs a family of strongly regular graphs and symmetric association schemes from complete graphs of specific sizes, providing explicit eigenmatrices and new algebraic structures.

## Contribution

It introduces a novel decomposition of complete graphs into strongly regular graphs, leading to explicit symmetric association schemes with determined eigenmatrices.

## Key findings

- Decomposition of complete graphs into strongly regular graphs for powers of two.
- Explicit eigenmatrices of the symmetric association schemes.
- Derivation of eigenmatrices for associated strongly regular graphs.

## Abstract

For any positive integer $m$, the complete graph on $2^{2m}(2^m+2)$ vertices is decomposed into $2^m+1$ commuting strongly regular graphs, which give rise to a symmetric association scheme of class $2^{m+2}-2$. Furthermore, the eigenmatrices of the symmetric association schemes are determined explicitly. As an application, the eigenmatrix of the commutative strongly regular decomposition obtained from the strongly regular graphs is derived.