Just a quick question here, is there a known description of a graph family where for every graph $G=(V,E)$ it holds that for every $(u,v) \in E$ you have $|N(u) \cap N(v)| \geq k$? There was a definition for such graph as $k$-community. Even non-trivial examples (e.g., cliques) might be interesting.
Also, how to construct a graph which is a $k$-community, but does not have a $k+2$ clique? Is it possible at all?
Note that non-trivial cases also exclude not connected graphs and 1-vertex graphs.