# Equitable 2-partitions of the Hamming graphs with the second eigenvalue

## TL;DR

This paper investigates equitable 2-partitions of Hamming graphs with a specific eigenvalue, extending previous characterizations and identifying new constructions beyond known cases.

## Contribution

It characterizes equitable 2-partitions of Hamming graphs with eigenvalue λ₂(n,q), generalizing prior results and discovering two new constructions.

## Key findings

- Equitable 2-partitions with eigenvalue λ₂(n,q) are reducible to those of H(3,q).
- Two new constructions of such partitions are identified.
- The study extends the understanding of spectral properties of Hamming graphs.

## Abstract

The eigenvalues of the Hamming graph $H(n,q)$ are known to be $\lambda_i(n,q)=(q-1)n-qi$, $0\leq i \leq n$. The characterization of equitable 2-partitions of the Hamming graphs $H(n,q)$ with eigenvalue $\lambda_{1}(n,q)$ was obtained by Meyerowitz in [15]. We study the equitable 2-partitions of $H(n,q)$ with eigenvalue $\lambda_{2}(n,q)$. We show that these partitions are reduced to equitable 2-partitions of $H(3,q)$ with eigenvalue $\lambda_{2}(3,q)$ with exception of two constructions.