  Let $A$ be a $d$-dimensional local ring containing a field. We will prove
that the highest Lyubeznik number $\lambda_{d,d}(A)$ (defined in \cite{l1}) is
equal to the number of connected components of the Hochster-Huneke graph
(defined in \cite{hh}) associated to $B$, where $B=\hat{\hat{A}^{sh}}$ is the
completion of the strict Henselization of the completion of $A$. This was
proven by Lyubeznik in characteristic $p>0$. Our statement and proof are
characteristic-free.
