Edge growth in graph powers

TL;DR

This paper establishes a lower bound on the number of edges in the rth power of a graph based on its order and minimal degree, providing insights into edge growth in graph powers.

Contribution

It introduces a new lower bound for edges in graph powers and characterizes minimal ratios for regular graphs with large diameter.

Findings

Derived a lower bound for e(G^r) in terms of graph order and minimal degree.

Determined minimal edge ratio e(G^r)/e(G) for regular graphs with diameter at least r.

Provides theoretical insights into the edge growth behavior in graph powers.

Abstract

For a graph G, its rth power G^r has the same vertex set as G, and has an edge between any two vertices within distance r of each other in G. We give a lower bound for the number of edges in the rth power of G in terms of the order of G and the minimal degree of G. As a corollary we determine how small the ratio e(G^r)/e(G) can be for regular graphs of diameter at least r.