  We relate two important notions in graph theory: expanders which are highly
connected graphs, and modularity a parameter of a graph that is primarily used
in community detection. More precisely, we show that a graph having modularity
bounded below 1 is equivalent to it having a large subgraph which is an
expander.
  We further show that a connected component $H$ will be split in an optimal
partition of the host graph $G$ if and only if the relative size of $H$ in $G$
is greater than an expansion constant of $H$. This is a further exploration of
the resolution limit known for modularity, and indeed recovers the bound that a
connected component $H$ in the host graph~$G$ will not be split
if~$e(H)<\sqrt{2e(G)}$.
