Structural Equivalence in Graphs and Complete Skeletons

Abstract

Two vertices $u$ and $v$ of a graph $\Gamma$ are strucuturally equivalent if and only if the transposition $(uv)$ 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 $\Gamma$.

18 pages, 7 figures