On nicely distance-balanced of folded cube graphs

Subject Areas : Statistics

Seyedeh Maryam Hosseini pour ¹ (Department of Pure Mathematics, Ferdowsi University of Mashhad,)
Freydoon Rahbarnia ² (Department of Applied Mathematics, Ferdowsi University ofMashhad,)
Mehdi Alaeiyan ³ (School of Mathematic, Iran University of Science and Technology, Tehran, Iran.)
Ahmad Erfanian ⁴ (Department of Mathematics and Center of Excellence in Analysis on Algebraic Structure)

Received: 2019-12-18 Accepted : 2020-04-19 Published : 2021-03-01

Keywords: گراف همینگ, خوش فاصله متوازن یالی, مکعب فولدد, ابرمکعب, خوش فاصله متوازن,

Abstract :

A nontrivial graph is called nicely distance-balanced (nicely edge distance-balanced), whenever there exist positive integers γ_V (γ_E), such that for any adjacent vertices u and v in V(Γ), there are exactly γ_V vertices in V(Γ) (γ_E edges in E(Γ) that are closer to u than v, and exactly γ_V vertices in V(Γ) (γ_E edges in E(Γ)) that are closer to v than u. In this paper, we will prove that hyper cube Q_n and the folded cube F_n are nicely distance-balanced and Q_n is also nicely edge distance-balanced.A nontrivial graph is called nicely distance-balanced (nicely edge distance-balanced), whenever there exist positive integers γ_V (γ_E), such that for any adjacent vertices u and v in V(Γ), there are exactly γ_V vertices in V(Γ) (γ_E edges in E(Γ) that are closer to u than v, and exactly γ_V vertices in V(Γ) (γ_E edges in E(Γ)) that are closer to v than u. In this paper, we will prove that hyper cube Q_n and the folded cube F_n are nicely distance-balanced and Q_n is also nicely edge distance-balanced.

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