  We present new values and bounds on the (normalised) closeness centrality
$\bar{\mathsf{C}}_C$ of connected graphs and on its product
$\bar{l}\bar{\mathsf{C}}_C$ with the mean distance $\bar{l}$ of these graphs.
Our main result presents the fundamental bounds $1\leq
\bar{l}\bar{\mathsf{C}}_C<2$. The lower bound is tight and the upper bound is
asymptotically tight. Combining the lower bound with known upper bounds on the
mean distance, we find ten new lower bounds for the closeness centrality of
graphs. We also present explicit expressions for $\bar{\mathsf{C}}_C$ and
$\bar{l}\bar{\mathsf{C}}_C$ for specific families of graphs. Elegantly and
perhaps surprisingly, the asymptotic values $n\bar{\mathsf{C}}_C\big(P_n\big)$
and of $n\bar{\mathsf{C}}_C\big(L_n\big)$ both equal $\pi$, and the asymptotic
limits of $\bar{l}\bar{\mathsf{C}}_C$ for these families of graphs are both
equal to $\pi/3$. We conjecture that the set of values
$\bar{l}\bar{\mathsf{C}}_C$ for all connected graphs is dense in the interval
$[1,2)$.
