If $Γ$ is a graph for which every edge is in exactly one clique of order $ω$, then one can form a new graph with vertex set equal to these cliques. This is a generalization of the line graph of $Γ$. We discover many general results and classifications related to these clique graph that will be useful to researchers studying these objects. In particular, we find bounds on its eigenvalues (with exact results when $Γ$ is $k$-regular) and some complete classifications when $Γ$ is strongly regular. We apply our results to many examples, including Conway's 99-graph problem and the existence problem for other strongly regular graphs.