On certain regular nicely distance-balanced graphs

Blas Fernández ^[1] ; Štefko Miklavič ^[1] ; Safet Penjić ^[1]
1. [1] University of Primorska
  
  University of Primorska
  
  Eslovenia
Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 65, Nº. 1, 2023, págs. 165-185
Idioma: inglés
DOI: 10.33044/revuma.2709
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Resumen
- A connected graph Γ is called nicely distance-balanced, whenever there exists a positive integer γ = γ(Γ) such that, for any two adjacent vertices u, v of Γ, there are exactly γ vertices of Γ which are closer to u than to v, and exactly γ vertices of Γ which are closer to v than to u. Let d denote the diameter of Γ. It is known that d ≤ γ, and that nicely distance-balanced graphs with γ = d are precisely complete graphs and cycles of length 2d or 2d + 1. In this paper we classify regular nicely distance-balanced graphs with γ = d + 1.