Equidistant dimension of Johnson and Kneser graphs

TL;DR

This paper explores the equidistant dimension of Johnson and Kneser graphs, establishing new exact values and bounds for specific graph classes, advancing understanding of their metric properties.

Contribution

It introduces properties of the equidistant dimension, derives exact values for certain Johnson and Kneser graphs, and provides bounds for others, expanding the theoretical framework.

Findings

Exact equidistant dimension for Johnson graphs J_{n,2}

Exact equidistant dimension for Kneser graphs K_{n,2}

Bounds for equidistant dimension of J_{n,3} and J_{2k,k}

Abstract

In this paper the recently introduced concept of equidistant dimension $eqdim(G)$ of graph $G$ is considered. Useful property of distance-equalizer set of arbitrary graph $G$ has been established. For Johnson graphs $J_{n,2}$ and Kneser graphs $K_{n,2}$ exact values for $eqdim(J_{n,2})$ and $eqdim(K_{n,2})$ have been derived, while for Johnson graphs $J_{n,3}$ it is proved that $eqdim(J_{n,3}) \le n-2$. Finally, exact value of $eqdim(J_{2k,k})$ for odd $k$ has been presented.