Consider graph $T$ where nodes correspond to maximal cliques of some graph $G$ and two nodes can be connected if corresponding cliques intersect. Clique tree is an example when $T$ is required to be a tree and $G$ is chordal. I'm interested in graphs $T$ when tree/chordal requirements are relaxed, do they come up anywhere?

Motivation: I come across these graphs when looking at approximate decompositions of Ising model entropy, searching for "maximal clique intersection graphs" only gives me literature related to clique trees/chordal graphs