Maximal $m$-distance sets containing the representation of the Johnson graph $J(n, m)$

TL;DR

This paper classifies the maximal m-distance sets in Euclidean space that include Johnson graph representations, providing conditions for their maximality and extending results to two-distance sets.

Contribution

It offers a complete classification of maximal m-distance sets containing Johnson graph representations for specific m values and establishes criteria for their maximality.

Findings

Classified maximal m-distance sets containing Johnson graph representations for m=2,3.

Determined necessary and sufficient conditions for maximality of Johnson graph representations.

Classified maximal two-distance sets containing J(n-1, 2).

Abstract

We classify the maximal $m$-distance sets in $\mathbb{R}^{n-1}$ which contain the representation of the Johnson graph $J(n, m)$ for $m = 2, 3$. Furthermore, we determine the necessary and sufficient condition for $n$ and $m$ such that the representation of the Johnson graph $J(n, m)$ is not maximal as an $m$-distance set. Also, we classify the maximal two-distance sets in $\mathbb{R}^{n-1}$ which contain the representation of $J(n - 1, 2)$.