How does one formally "denote" a countably infinite family of finite graphs? I'd be tempted to write:
$\Gamma: \mathbb{N}\to \mathbb{N}\times 2^{\mathbb{N}^2}$
but it feels "unwieldy" and not quite right (among other things, it "allows" countably infinite edges). However, I'm looking for something more formal than $\{G_1,G_2,\dots\}$. Any "standard" way to do it succintly?
Succinctly is the key. I can obviously give a formal definition of what a graph is, what an infinite family of objects is etc. But the point is that I want to denote a family of finite graphs as I write: "Consider a countably infinite family of graphs $\Gamma: \{G_1,G_2,\dots\}$; and let $C_\Gamma$ be ..."
If $\mathcal G$ is the class of all graphs of interest, then you want $\Gamma\colon \omega\to \mathcal G$. Now formalize $\mathcal G$.
Given a set $S$, let $E(S)=\{\,e\in 2^S:|e|=2\,\}.$ Then a simple, undirected graph on vertex set $V$ can be represented as a pair $\langle V,E\rangle$ with $E\subseteq E(V)$. Thus the class of all finite, simple, undirected graphs would be $$\mathcal G=\{\,\langle V,E\rangle: |V|<\infty,E\in2^{2^V},\forall e\in E\colon|e|=2\,\}$$