🤖 AI Summary
This work establishes a prime-counting theory within graph homology classes, formulating a graph-theoretic analogue of the Dirichlet prime number theorem. Addressing the distribution of graph primes—i.e., irreducible closed walks—across homology classes, it introduces twisted adjacency matrices and conducts spectral analysis over the character group of the first homology. It rigorously proves that the associated spectrum is anti-symmetric about the origin and identifies the canonical character as yielding extremal spectral values. Building upon group representation theory and homological analysis, the paper develops a novel trace formula framework, yielding several exact trace formulas. This constitutes the first homologically refined characterization of prime distribution on graphs, bridging spectral graph theory, arithmetic geometry, and combinatorial topology through a unified paradigm.
📝 Abstract
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.