I am looking for a non-complete regular graph for which the two following conditions hold:
- for each two adjacent vertices $u,v$, $\left|N(u)\cap N(v)\right|=2$
- for each two non-adjacent vertices $u,v$, $\left|N(u)\cap N(v)\right|=1$
where $N(x)$ is the set of neighbours of $x$ (excluding $x$ itself).
I'm ok with replacing 2 with 3, say (or actually, with any positive constant...). Played a little bit with the pencil and with the computer, and couldn't find such.
Looking at this page, I found three examples on fewer than 500 vertices: srg(209,16,3,1), srg(375,16,3,1), and srg(400,21,2,1). I have no idea how these graphs were constructed - perhaps you can contact the author of that page.
Edit: Actually, if I'm understanding the color notation of the rows on the first site (green=provably exists, yellow=potentially exists, red=provably doesn't exist), the only one of those that could exist is the one with 400 vertices. I found three more examples of feasible parameters with $\mu=1$, but they were all colored red.