Bounds on the closeness centrality of a graph

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 $[1,2)$.