Growth of graph powers

TL;DR

This paper investigates how the number of edges in a graph increases when constructing its rth power, showing that either the power is complete or many new edges are added, extending previous results on graph cubes.

Contribution

It extends prior work by P. Hegarty on graph cubes to general rth powers, providing new insights into the edge growth in graph powers.

Findings

Either the rth power of a graph is complete or many new edges are added.

Provides bounds on the number of edges added in graph powers.

Generalizes previous results on graph cubes to rth powers.

Abstract

For a graph G, its rth power is constructed by placing an edge between two vertices if they are within distance r of each other. In this note we study the amount of edges added to a graph by taking its rth power. In particular we obtain that either the rth power is complete or "many" new edges are added. This is an extension of a result obtained by P. Hegarty for cubes of graphs.