If $G=\left(V\right(G),E(G\left)\right)$ is a simple connected graph with the vertex set $V\left(G\right)$ and the edge set $E\left(G\right)$ , S is a subset of $V\left(G\right)$ , and let $B\left(S\right)$ be the set of neighbors of S in $V\left(G\right)\setminus S$ . Then, the differential of S $\partial \left(S\right)$ is defined as $\left|B\right(S\left)\right|-\left|S\right|$ . The differential of G, denoted by $\partial \left(G\right)$ , is the maximum value of $\partial \left(S\right)$ for all subsets $S\subseteq V\left(G\right)$ . The graph operator $Q\left(G\right)$ is defined as the graph that results by subdividing every edge of G once and joining pairs of these new vertices iff their corresponding edges are incident in G. In this paper, we study the relations between $\partial \left(G\right)$ and $\partial \left(Q\left(G\right)\right)$ . Besides, we exhibit some results relating the differential $\partial \left(G\right)$ and well-known graph invariants, such as the domination number, the independence number, and the vertex-cover number.

Let P be a set of $n\ge 3$ points in general position in the plane. The edge disjointness graph $D\left(P\right)$ of P is the graph whose vertices are the $\left(\genfrac{}{}{0pt}{}{n}{2}\right)$ closed straight line segments with endpoints in P, two of which are adjacent in $D\left(P\right)$ if and only if they are disjoint. In this paper we show that the connectivity of $D\left(P\right)$ is at most $\frac{7n^2}{18}+\Theta \left(n\right)$, and that this upper bound is asymptotically tight. The proof is based on the analysis of the connectivity of $D\left(Q_n\right)$, where $Q_n$ denotes an n-point set that is almost 3-symmetric.