  Let $G=(V(G), E(G))$ be a simple graph with vertex set $V(G)$ and edge set
$E(G)$. Let $S$ be a subset of $V(G)$, and let $B(S)$ be the set of neighbours
of $S$ in $V(G) \setminus S$. The differential $\partial(S)$ of $S$ is the
number $|B(S)|-|S|$. The maximum value of $\partial(S)$ taken over all subsets
$S\subseteq V(G)$ is the differential $\partial(G)$ of $G$. The graph $R{G}$ is
defined as the graph obtained from $G$ by adding a new vertex $v_e$ for each
$e\in E(G)$, and by joining $v_e$ to the end vertices of $e$. In this paper we
study the relationship between $\partial(G)$ and $\partial(R(G))$, and give
tight asymptotic bounds for $\partial(R(G))$. We also exhibit some
relationships between certain vertex sets of $G$ and $R(G)$ which involve well
known graph theoretical parameters.
