  A prominent parameter in the context of network analysis, originally proposed
by Watts and Strogatz (Collective dynamics of `small-world' networks, Nature
393 (1998) 440-442), is the clustering coefficient of a graph $G$. It is
defined as the arithmetic mean of the clustering coefficients of its vertices,
where the clustering coefficient of a vertex $u$ of $G$ is the relative density
$m(G[N_G(u)])/{d_G(u)\choose 2}$ of its neighborhood if $d_G(u)$ is at least
$2$, and $0$ otherwise. It is unknown which graphs maximize the clustering
coefficient among all connected graphs of given order and size.
  We determine the maximum clustering coefficients among all connected regular
graphs of a given order, as well as among all connected subcubic graphs of a
given order. In both cases, we characterize all extremal graphs. Furthermore,
we determine the maximum increase of the clustering coefficient caused by
adding a single edge.
