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

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 distanceregular 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. 2020 Mathematics Subject Classification. 05C69 and 05E30.