TITLE: "Common-neighbor-regular" graphs
QUESTION [5 upvotes]: Let $G=(V,E)$ be a finite, simple, undirected graph. For $v\in V$ we set $N(v)=\{w\in V:\{v,w\}\in E\}$.
We say that $G$ is $k$-common-neighbor-regular if for all $v\neq w\in V$ we have $|N(v)\cap N(w)|=k$. (I haven't been able to find out whether there is a canonical term for this concept.)
Clearly the complete graph on $k+2$ points is $k$-common-neighbor-regular, but for $k>0$ I haven't been able to find non-complete examples.
Question: Given $k>0$, for which $n\in\mathbb{N}$ does there exist a $k$-common-neighbor-regular graph $G=(V,E)$ with $|V|=n$?

REPLY [2 votes]: This problem was (partially?) solved by Kelmans in 1973 and by Ralucca Gera and Jian Shen in 2008.  See this paper including the comment. They consider the generalization that friends have $\lambda$ common friends and non-friends have $\mu$ common friends. If the graph is regular, it is strongly-regular by definition. If it is irregular, it is either a disjoint union of complete graphs of some specified sizes, or a disjoint union of complete graphs of the same size plus an extra vertex joined to everything.