  The concept of the \emph{Gutman index}, denoted $Gut(G)$ was introduced for a
connected undirected graph $G$. In this note we apply the concept to the
underlying graphs of the family of Jaco graphs, (\emph{directed graphs by
definition}), and describe a recursive formula for the \emph{Gutman index}
$Gut(J^*_{n+1}(x)).$ We also determine the \emph{Gutman index} for the trivial
\emph{edge-joint} between Jaco graphs.
