An edge clique cover of an undirected graph $G$ is a set of cliques of $G$ such that every edge of $G$ is an edge in at least one clique in the set. The edge clique cover number $\theta(G)$ is the minimum number of cliques in an edge clique cover of $G$. Equivalently, this is the intersection number.
Erdös, Goodman, and Posa  have shown that $\theta(G) \leq \lfloor n^2/4 \rfloor$. Moreover, there is a covering with $\lfloor n^2/4 \rfloor$ edges and triangles. For interval graphs, $\theta(G)$ equals the number of maximal cliques of $G$ (minus the number of isolated vertices of $G$) .
Are there tighter upper bounds known for other special classes of graphs? There is a survey by Fred Roberts , which states: "... there has recently been a large amount of work on these coverings, much of which is in widely scattered places in the literature. It is remarkable how similar these widely scattered results are." The survey is from 1985, and I don't see it mentioning results for classes other than interval graphs. Maybe there are newer developments (that don't cite the survey, and thus I'm not finding them)?
 Opsut, Robert J., and Fred S. Roberts. "On the fleet maintenance, mobile radio frequency, task assignment, and traffic phasing problems." The theory and applications of graphs (1981): 479-492.