Substructure Analysis and Cycle Enumeration Methods for Oriented Graphs Based on Parameterizing Hermitian Laplacian Matrices by Galois Conjugates

This paper investigates the principal minors of a parameterized Hermitian Laplacian matrix for oriented graphs. Particularly, we focus on the properties of the matrix for parameters chosen as Galois conjugates of a primitive $p$th root of unity, where $p$ is an odd prime. We demonstrate that under this condition, the product of the corresponding Hermitian Laplacian determinants is an integer power of $p$. This algebraic property forms the basis for a method to enumerate non-vanishing unicyclic graph components within certain substructures. The study is situated within a framework where a variable unit-modulus complex parameter is introduced into the Hermitian Laplacian matrix, which also allows for an examination of relationships among principal minors under different parameters. Our analysis adopts the concept of substructures, defined as vertex-edge pairs $(V',E')$ where edges in $E'$ are not restricted to connecting vertices within $V'$.