The differential on Graph Operator R(G)

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 $\mathrm{\partial }(S)$ of $S$ is the number $\mathrm{\mid }B(S)\mathrm{\mid }-\mathrm{\mid }S\mathrm{\mid }$. The maximum value of $\mathrm{\partial }(S)$ taken over all subsets $S\subseteq V(G)$ is the differential $\mathrm{\partial }(G)$ of $G$. The graph $RG$ 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 $\mathrm{\partial }(G)$ and $\mathrm{\partial }(R(G))$, and give tight asymptotic bounds for $\mathrm{\partial }(R(G))$. We also exhibit some relationships between certain vertex sets of $G$ and $R(G)$ which involve well known graph theoretical parameters.