2024/12/25 by Gang Ma, Ma, Gang, Jianfeng Wang +5
Mathematics · #Finite Group Theory Research #Mathematics and Applications #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2412.18893
A connected graph Γ of diameter \rm diam(Γ) ≥ ℓ is ℓ-distance-balanced if |Wxy(Γ)|=|Wyx(Γ)| for every x,y∈ V(Γ) with dΓ(x,y)=ℓ, where Wxy(Γ) is the set of vertices of Γ that are closer to x than to y. Γ is said to be highly distance-balanced if it is ℓ-distance-balanced for every ℓ∈ [\rm diam(Γ)]. It is proved that every cubic Cayley graph whose generating set is one of \a,an-1,bar\ and \ak,an-k,bat\ is highly distance-balanced. This partially solves a problem posed by Miklavič and Šparl.