Maximal clique intersection graphs 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
 A: $T$ is called the clique graph of $G$, see
https://link.springer.com/chapter/10.1007/0-387-22444-0_5
A: Hi,
I have been working with clique graphs for the last 10 years. My doctoral tesis was about
clique graphs. Recently we have proved that recognizing clique graphs is an NP-complete problem. 
See   Theoretical Computer Science
Volume 410, Issues 21-23, 17 May 2009, Pages 2072-2083 
What exactly do you need to know about clique graphs?
liliana
A: Hi,
It is not an answer, just a comment. This site may be relevant: http://www.eprisner.de/Journey/CliqueGraphs.html 
Everything changes drastically if you manage to use not clique intersection but clique incidence matrix in your decomposition. Then you immediately fall into the realm of perfect graphs.
The rows of clique-incidence matrix $A$ of a graph $G$ are incidence vectors of (maximal) cliques and vertices. For perfect graphs (that are graphs not containing induced odd cycles of the length greater than three or their complement) these matrices have several nice properties with respect to packing and covering the vertices by the subsets --- maximal cliques. Formally, the polytope $\{x : Ax\leq e, x\geq 0\}$ ($e$ is a vector of all $1$') is integral iff $G$ is perfect. In particular, a lot of NP-hard in general problems (e.g. chromatic number) are polynomial for perfect graphs. But coumting is a more subtle matter.
