Spectral Antisymmetry of Twisted Graph Adjacency

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.