Structural Equivalence in Graphs and Complete Skeletons

TL;DR

This paper explores the concept of structural equivalence in graphs, introduces the complete skeletons of graphs, and uses these to analyze automorphism groups and eigenvalue multiplicities, providing new tools for graph analysis.

Contribution

It develops the notion of complete skeletons of graphs based on structural equivalence and relates them to automorphism groups and eigenvalue properties, offering new analytical methods.

Findings

Characterization of structural equivalence and automorphism groups.

Introduction of complete skeletons and their relation to compression graphs.

Formula for rank(I+A(Γ)) using complete skeletons.

Abstract

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 $\Gamma$.