Two vertices $u$ and $v$ of a graph $Γ$ are strucuturally equivalent if and only if the transposition $(u\,v)$ is in Aut($Γ$), the automorphism group of $Γ$. Some properties of structural equivalence and the group of vertex permutations generated by the transpositions in Aut($Γ$) 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 $Γ$, denoted $Ω(Γ)$, is used to find a formula for rank$(I+A(Γ))$, which is helpful for determining the multiplicity of the -1 eigenvalue of $Γ$.