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. 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，提出了素数定理的一种图论狄利克雷型类比。主要创新在于建立并应用了谱反对称性定理，该定理揭示了扭转图邻接矩阵的谱在图特征群上具有反对称分布。此外，作为分析的一部分，我们还基于扭转邻接矩阵推导了一些迹公式。