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.