Clique graph of bipartite graphs The clique graph $C$ of a given graph $G$ has the maximal cliques of $G$ as vertices and their is an edge between two vertices in $C$ iff the corresponding cliques share some vertices.

Now for chordal graphs, this clique graph is a tree and for proper interval graphs it is a path. Incidentally for both these graphs the max-clique recognition algorithm runs in polynomial time.

My question is are there other class of graphs who has characterization in terms of its clique graph. ? I am specially looking for such characterisations for bipartite graphs. 

Any link to paper/journal is welcome.
I recommend you taking a look at this book chapter:


J. L. Szwarcfiter, A Survey on Clique Graphs, Recent Advances in Algorithms and Combinatorics,
CMS Books in Mathematics 2003, pp 109-136.


In there, many characterizations of clique graphs of specific classes are given. For example, it is mentioned that the clique graphs of chordal graphs are the dually chordal graphs and clique graphs of proper interval graphs are also proper interval graphs, and not trees and paths as you supposed.