On convergence structures in graphs

A closure operator on a set $X$ is a function $\operatorname{cl}: \wp(X) \to \wp(X)$ satisfying, for all $A, B \subseteq X$, the following properties: extensivity, $A \subseteq \operatorname{cl}(A)$; monotonicity, which states that if $A \subseteq B$ then $\operatorname{cl}(A) \subseteq \operatorname{cl}(B)$; and preservation of unions, $\operatorname{cl}(A \cup B) = \operatorname{cl}(A) \cup \operatorname{cl}(B)$. Every graph $G$ naturally carries such an operator on its vertex set by assigning to each subset $A \subseteq V(G)$ the set $\operatorname{cl}(A) = A \cup N(A)$, where $N(A)$ denotes the vertices adjacent to a vertex in $A$. Since closure operators and pretopological spaces are equivalent notions, this operator induces a canonical convergence structure on $V(G)$. We describe this convergence in terms of nets and relate combinatorial properties of the graph to convergence-theoretic ones.