Edge growth in graph squares

TL;DR

This paper proves a lower bound on the number of edges in the square of a regular graph, confirming a conjecture and revealing new insights into graph structure and edge growth.

Contribution

It resolves Hegarty's conjecture by establishing a minimum edge increase in the square of regular graphs, except for specific narrow families.

Findings

The square of a connected regular graph typically has significantly more edges.

The result applies to all but two narrow families of graphs.

The paper confirms a conjecture about edge growth in graph squares.

Abstract

We resolve a conjecture of Hegarty regarding the number of edges in the square of a regular graph. If $G$ is a connected $d$-regular graph with $n$ vertices, the graph square of $G$ is not complete, and $G$ is not a member of two narrow families of graphs, then the square of $G$ has at least $(2-o_d(1))n$ more edges than $G$.