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

Structural Equivalence in Graphs and Complete Skeletons