Distance-regular graphs admitting a perfect $1$-code

TL;DR

This paper investigates which distance-regular graphs can contain a perfect 1-code, providing characterizations for specific classes such as line graphs, small valency graphs, and graphs with certain girth and valency conditions.

Contribution

It offers new characterizations of distance-regular graphs that admit perfect 1-codes, including line graphs and graphs with specific valency and girth properties.

Findings

Characterization of distance-regular line graphs with perfect 1-codes.

Identification of all known small valency distance-regular graphs with perfect 1-codes.

Analysis of graphs with girth 3 and valency 6 or 7 admitting perfect 1-codes.

Abstract

In this paper, we study the problem that which of distance-regular graphs admit a perfect $1$-code. Among other results, we characterize distance-regular line graphs which admit a perfect $1$-code. Moreover, we characterize all known distance-regular graphs with small valency at most $4$, the distance-regular graphs with known putative intersection arrays for valency $5$, and all distance-regular graphs with girth $3$ and valency $6$ or $7$ which admit a perfect $1$-code.