Edge Growth in Graph Cubes.

link: http://arxiv.org/abs/1009.0343
Abstract

We show that for every connected graph $G$ of diameter $\ge 3$, the graph
$G^3$ has average degree $\ge 7/4 \delta(G)$. We also provide an example
showing that this bound is best possible. This resolves a question of Hegarty
\cite{PH}.