I'm looking for any prior work on a (for lack of a better word) meta-graph. Let $M=(V,E)$ be a a meta-graph with vertex and edge set $V,E$. The meta-graph is formed by mapping a set of graphs $G = \{g_1, g_2, \ldots \}$ onto the vertices of $V$, one-to-one $g_1 \rightarrow v_1, g_2 \rightarrow v_2, \ldots$ and there is an edge between $v_i, v_j \in V$ if $g_i$ and $g_J$ share some property.
Example #1: For example, let $G$ be the set of valid $k$ colorings over the complete graph $K_n$ and let the edge function be a single color change, e.g. $v_i$ and $v_j$ share an edge if a single color change in $g_i$ would result in $g_j$.
Example #2: Let $G$ include $K_n$ and all of it's subgraphs. The edge function is such that $v_i$ and $v_j$ share an edge if the number of edges and vertices $g_i$ is exactly one less than $g_j$.
The question is a reference request. I'd like to know the proper name for the "meta-graph" and what kinds have been studied.