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 graphs that will be useful to researchers studying graphs with this property. In particular, we find bounds on the spectrum of $Γ$ (with exact results when $Γ$ is $k$-regular) and some complete classifications when $Γ$ is strongly regular. We apply our results to derive novel information about the existence questions of certain strongly regular graphs. We also examine the critical group of graphs with this property and their associated transformations. Finally, we discuss examples of widely studied families of graphs that have this property, and provide some examples to make the results more concrete.