标题： 基于伽罗瓦共轭参数化厄米特拉普拉斯矩阵的有向图子结构分析与环枚举方法 显示英文标题

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

摘要： 本文研究了有向图的参数化厄米特拉普拉斯矩阵的主子式。 特别地，我们关注当参数选择为原始$p$次单位根的伽罗瓦共轭时该矩阵的性质，其中$p$是一个奇素数。 我们证明在这一条件下，相应的厄米特拉普拉斯行列式的乘积是$p$的整数次幂。 这一代数性质构成了在某些子结构中枚举非零单环图成分的方法的基础。 本研究处于一个框架内，其中引入了一个变量模长复参数到厄米特拉普拉斯矩阵中，这也允许对不同参数下主子式之间的关系进行考察。 我们的分析采用了子结构的概念，定义为顶点-边对$(V',E')$，其中$E'$中的边不限于连接$V'$内的顶点。 显示英文摘要

摘要： 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 $V'$.