# Weighted prime geodesic theorems

## TL;DR

This paper extends prime geodesic theorems to weighted infinite graphs and building quotients, relating growth rates to spectral data of translation operators, inspired by Bass's work.

## Contribution

It introduces weighted prime geodesic theorems for complex structures, connecting spectral analysis with geometric growth in new weighted settings.

## Key findings

- Prime geodesic theorems are established for weighted graphs.
- Growth rates are expressed via spectral data of translation operators.
- The approach generalizes classical results to weighted and more complex structures.

## Abstract

Prime geodesic theorems for weighted infinite graphs and weighted building quotients are given. The growth rates are expressed in terms of the spectral data of suitable translation operators inspired by a paper of Bass.