2016/02/03 by Saori Adachi, Adachi, Saori, Rina Hayashi +5 · 1 citation
Computer Science · Mathematics · #05B30 #05D05 #Combinatorics (math.CO) #Digital Image Processing Techniques #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.1602.01215
openalex publication_date 2016/02/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
A set X in the Euclidean space ℝd is called an m-distance set if the set of Euclidean distances between two distinct points in X has size m. An m-distance set X in ℝd is said to be maximal if there does not exist a vector x in ℝd such that the union of X and \x\ still has only m distances. Bannai--Sato--Shigezumi (2012) investigated the maximal m-distance sets which contain the Euclidean representation of the Johnson graph J(n,m). In this paper, we consider the same problem for the Hamming graph H(n,m). The Euclidean representation of H(n,m) is an m-distance set in ℝm(n-1). We prove that the maximum n is m2 + m - 1 such that the representation of H(n,m) is not maximal as an m-distance set. Moreover we classify the largest m-distance sets which contain the representation of H(n,m) for m≤ 4 and any n. We also classify the maximal 2-distance sets in ℝ2n-1 which contain the representation of H(n,2) for any n.