If I use $(W(E), V, E)$ to denote a graph with $E$ set of edges $V$ set of vertices and edge weight $W(E)$. Then, will my notations ($\emptyset, \infty, \emptyset$) and ($\emptyset, \emptyset, \emptyset$) be appropriate to denote an empty graph and a null graph, respectively? Where empty graph is defined as the graph with no edges and no vertices and null graph is defined as a graph with no edges as defined by an expert here.
The infinity symbol plays something of a different role in graph theory and I would not intuitively expect it to denote an arbitrary set of vertices. This confusion is compounded by the fact that you already have a symbol for an arbitrary vertex set $V$. So unless your use of $\infty$ were clarified I would consider this poor notation.