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Distance-balancing of cube-connected cycles graphs

Mokhtar Aouina, Hamed Karami, Douglas J. LaFountain

Abstract

We show that cube-connected cycles graphs $CCC_n$ are distance-balanced, and nicely distance-balanced if and only if $n$ is even.

Distance-balancing of cube-connected cycles graphs

Abstract

We show that cube-connected cycles graphs are distance-balanced, and nicely distance-balanced if and only if is even.
Paper Structure (4 sections, 8 theorems, 5 equations, 3 figures)

This paper contains 4 sections, 8 theorems, 5 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Distance-balancing in $CCC_n$
  3. Interval-with-ends diagrams for paths in $CCC_n$
  4. Nice distance-balancing in $CCC_n$

Key Result

Lemma 2.1

A choice of vertex $u$ in $Q_n$ as $00\cdots0$, along with a choice of labeling for the $n$ vertices adjacent to $u$, determines the labeling of all vertices in $Q_n$.

Figures (3)

  • Figure 1: Four examples of interval-with-ends (IWE) diagrams from $00000001$ to $01011006$ in $CCC_7$. Part (a) has length 8, since three circles are included in the interval, and five edges in the loop are included in the interval-with-ends; part (b) has length 13; part (c) has length 11; and part (d) has length 16.
  • Figure 2: An interval $I$ for a possible IWE diagram realizing $d(x,u)$.
  • Figure 3: Intervals $I$ and $I'$ for possible IWE diagrams realizing $d(x,u)$ and $d(x,v)$, respectively.

Theorems & Definitions (10)

  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 4.1
  • Remark 4.2
  • Proposition 4.3
  • Theorem 4.4
  • Remark 4.5