  Let $S$ be a set of transpositions generating the symmetric group $S_n$. The
transposition graph of $S$ is defined to be the graph with vertex set
$\{1,\ldots,n\}$, and with vertices $i$ and $j$ being adjacent in $T(S)$
whenever $(i,j) \in S$. In the present note, it is proved that two
transposition graphs are isomorphic if and only if the corresponding two Cayley
graphs are isomorphic. It is also proved that the transposition graph $T(S)$ is
edge-transitive if and only if the Cayley graph $Cay(S_n,S)$ is
edge-transitive.
