  Two vertices $u$ and $v$ of a graph $\Gamma$ are strucuturally equivalent if
and only if the transposition $(u\,v)$ is in Aut($\Gamma$), the automorphism
group of $\Gamma$. Some properties of structural equivalence and the group of
vertex permutations generated by the transpositions in Aut($\Gamma$) are
discussed, along with the prime graphs of these groups. The notion of
structural equivalence is used to develop a way of reconfiguring graphs into
what are called their complete skeletons, which is closely related to
compression graphs. Finally, the complete skeleton of a graph $\Gamma$, denoted
$\Omega(\Gamma)$, is used to find a formula for rank$(I+A(\Gamma))$, which is
helpful for determining the multiplicity of the -1 eigenvalue of $\Gamma$.
