  This paper concerns $(r,c)$-constant graphs, which are $r$-regular graphs in
which the subgraph induced by the open neighbourhood of every vertex has
precisely $c$ edges. The family of $(r,c)$-graphs contains vertex-transitive
graphs (and in particular Cayley graphs), graphs with constant link (sometimes
called locally isomorphic graphs), $(r,b)$-regular graphs, strongly regular
graphs, and much more.
  This family was recently introduced in [arXiv:2312.08777] serving as
important tool in constructing flip graphs [arXiv:2312.08777,
arXiv:2401.02315].
  In this paper we shall mainly deal with the following:
  i. Existence and non-existence of $(r, c)$-planar graphs. We completely
determine the cases of existence and non-existence of such graphs and supply
the smallest order in the case when they exist.
  ii. We consider the existence of $(r, c)$-circulant graphs. We prove that for
$c \equiv 2 \ (\mathrm{mod} \ 3)$ no $(r,c)$-circulant graph exists and that
for $c \equiv 0, 1 \ (\mathrm{mod} \ 3)$, $c > 0$ and $r \geq 6 +
\sqrt{\frac{8c - 5}{3}}$ there exists $(r,c)$-circulant graphs. Moreover for $c
= 0$ and $r \geq 1$, $(r, 0)$-circulants exist.
  iii. We consider the existence and non-existence of small $(r,c)$-constant
graphs, supplying a complete table of the smallest order of graphs we found for
$0 \leq c \leq \binom{r}{2}$ and $r \leq 6$. We shall also determine all the
cases in this range for which $(r,c)$-constant graphs don't exist. We establish
a public database of $(r,c)$-constant graphs for varying $r$, $c$ and order.
