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Proximity and Remoteness in Graphs: a survey

Mustapha Aouchiche, Bilal Ahmad Rather

Abstract

The proximity $π= π(G)$ of a connected graph $G$ is the minimum, over all vertices, of the average distance from a vertex to all others. Similarly, the maximum is called the remoteness and denoted by $ρ= ρ(G)$. The concepts of proximity and remoteness, first defined in 2006, attracted the attention of several researchers in Graph Theory. Their investigation led to a considerable number of publications. In this paper, we present a survey of the research work.

Proximity and Remoteness in Graphs: a survey

Abstract

The proximity of a connected graph is the minimum, over all vertices, of the average distance from a vertex to all others. Similarly, the maximum is called the remoteness and denoted by . The concepts of proximity and remoteness, first defined in 2006, attracted the attention of several researchers in Graph Theory. Their investigation led to a considerable number of publications. In this paper, we present a survey of the research work.
Paper Structure (5 sections, 145 theorems, 158 equations, 11 figures)

This paper contains 5 sections, 145 theorems, 158 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Proximity and Remoteness
  3. Proximity and Remoteness Compared to other Metric Invariants
  4. Proximity and Remoteness Compared to other Invariants
  5. Conclusion

Key Result

Proposition 2.1

Let $G$ be a connected graph on $n \ge 3$ vertices with proximity $\pi$ and remoteness $\rho$. Then and The lower bound on $\pi$ is reached if and only if $G$ contains a dominating vertex; the upper bound on $\pi$ is attained if and only if $G$ is either the cycle $C_n$ or the path $P_n$; the lower bound on $\rho$ is reached if and only if $G$ is the complete graph $K_n$; the upper bound on $\rh

Figures (11)

  • Figure 1: The transmission regular but not degree regular graph with the smallest order
  • Figure 2: Relations between the invariants.
  • Figure 3: The double-tailed comet $DTC_{15,4,4}$.
  • Figure 4: Graphs with $D=3$ that maximize $\rho + \overline{\rho}$ for $n=6$.
  • Figure 5: Graphs in family $\mathcal{A}$
  • ...and 6 more figures

Theorems & Definitions (146)

  • Proposition 2.1: Aouchiche2011
  • Proposition 2.2: Aouchiche2011
  • Proposition 2.3: Aouchiche2011
  • Proposition 2.4: Aouchiche2011
  • Theorem 2.5: Aouchiche2010
  • Theorem 2.6: Aouchiche2010
  • Theorem 2.7: Aouchiche2010
  • Theorem 2.8: Aouchiche2010
  • Theorem 2.9: czabarka2020
  • Lemma 2.10: payen
  • ...and 136 more