On completely regular codes with minimum eigenvalue in geometric graphs

TL;DR

This paper characterizes completely regular codes with minimum eigenvalue in geometric graphs, linking them to clique graphs, and fully classifies such codes in Johnson graphs J(n,3) and most in J(n,4).

Contribution

It provides a complete characterization of completely regular codes in Johnson graphs J(n,w) with specific parameters, extending previous classifications.

Findings

Characterization of completely regular codes in Johnson graphs J(n,w) with covering radius w-1 and strength 1.

Complete classification of such codes in J(n,3).

Most codes classified in J(n,4) with one eigenvalue case remaining open.

Abstract

We prove that any completely regular code with minimum eigenvalue in any geometric graph G corresponds to a completely regular code in the clique graph of G. Studying the interrelation of these codes, a complete characterization of the completely regular codes in the Johnson graphs J(n,w) with covering radius w-1 and strength 1 is obtained. In particular this result finishes a characterization of the completely regular codes in the Johnson graphs J(n,3). We also classify the completely regular codes of strength 1 in the Johnson graphs J(n,4) with only one case for the eigenvalues left open.