Substructure Analysis and Cycle Enumeration Methods for Directed Graphs Based on a Parameterized Hermitian Laplacian Matrix

Abstract: This paper investigates the principal minors of a parameterized Hermitian Laplacian matrix for directed graphs, with a particular focus on the properties that appear when its complex parameter is a primitive $5$th root of unity. We demonstrate that under this condition, the principal minors yield values within the quadratic field $\mathbb{Q}(\sqrt{5})$. This algebraic property forms the basis for a method to enumerate non-vanishing unicyclic graphs of certain substructures by analyzing the rational and irrational components of the Hermitian Laplacian determinant. The study is situated within a general framework where a variable unit-modulus complex parameter is introduced into the Hermitian Laplacian matrix, which also facilitates an examination of relationships between determinants for different roots of unity. Our analysis adopts and investigates the concept of substructures, defined as vertex-edge pairs $(V',E')$ where edges in $E'$ are not restricted to connecting vertices within $V'$.