We address a prime counting problem across the homology classes of a graph, presenting a graph-theoretical Dirichlet-type analogue of the prime number theorem. The main machinery we have developed and employed is a spectral antisymmetry theorem, revealing that the spectra of the twisted graph adjacency matrices have an antisymmetric distribution over the character group of the graph with a special character called the canonical character being an extremum. Additionally, we derive some trace formulas based on the twisted adjacency matrices as part of our analysis.

翻译：我们研究了图同调类上的素数计数问题，提出了图论中狄利克雷型素数定理的类比。我们所发展并运用的主要工具是一个谱反对称定理，该定理揭示了扭曲图邻接矩阵的谱在图的特征群上具有反对称分布，其中被称为典范特征的特殊特征是一个极值点。此外，作为分析的一部分，我们基于扭曲邻接矩阵推导出一些迹公式。