  For a graph property $\mathcal{P}$ and a common vertex set $V = \{1, 2,
\ldots, n\}$, a family of graphs on $V$ is \emph{$\mathcal{P}$-intersecting}
iff $G \cap H$ satisfies $\mathcal{P}$ for all $G,H$ in the family. Addressing
a question of Chung, Graham, Frankl, and Shearer, we explore---for various
$\mathcal{P}$---the maximum cardinality among all $\mathcal{P}$-intersecting
families of graphs. In the connected-intersecting case, we resolve the question
completely by a short linear algebraic proof showing this maximum is attained
by taking all graphs containing a fixed spanning tree (though we show other
extremal constructions as well). We also present a new lower bound for
containing unions of a fixed subgraph.
