A Turan-type problem on distances in graphs

TL;DR

This paper introduces a new problem related to distances in graphs, proposes conjectures, and provides bounds on the number of vertex pairs at a certain distance for large graphs.

Contribution

It formulates a novel Turan-type problem on graph distances and establishes initial bounds supporting the conjectures.

Findings

Graphs with no three vertices at pairwise distance k have at most (n-k+1)^2/4 pairs at distance k for large n and k.

Proposes several conjectures about distances in graphs.

Lays groundwork for future proofs of the conjectures.

Abstract

We suggest a new type of problem about distances in graphs and make several conjectures. As a first step towards proving them, we show that for sufficiently large values of n and k, a graph on n vertices that has no three vertices at pairwise distance k has at most (n-k+1)^2/4 pairs of vertices at distance k.