  For any natural number n, the Hurwitz graph $\mathcal{H}(S_n )$ has vertices
corresponding to reduced words of the cycle $(1 . . . n)$ in terms of
transpositions in $S_n$, and has edges corresponding to local shifts. This
graph is known to be connected, and some of its graph theoretic properties,
such as its radius and upper bounds on its diameter, are known. We explore
further the structure of the Hurwitz Graph, in particular its center and its
connection to the geometric tree graph $G_n$. Finally, the extended Hurwitz
Graph is defined and its properties such as the radius, characterization of
some central elements and bounds on the diameter are obtained. Part of
fulfillment of the requirements for the Master's Degree.
