A tower of Ramanujan graphs and a reciprocity law of graph zeta functions

TL;DR

This paper constructs a tower of Ramanujan graphs from modular curves, demonstrating bounded Cheeger constants, and establishes a reciprocity law for graph zeta functions, advancing understanding of graph spectra and zeta function symmetries.

Contribution

It introduces a novel construction of Ramanujan graph towers from modular curves and proves a reciprocity law for their zeta functions, linking graph theory and number theory.

Findings

Constructed a tower of Ramanujan graphs with degree l+1 from modular curves.

Demonstrated that the Cheeger constants of these graphs are bounded below by a specific value.

Proved a reciprocity law for graph (Ihara) zeta functions.

Abstract

Let l be an odd prime. We will construct a tower of connected regular Ramanujan graph of degree l+1 from of modular curves. This supplies an example of a collection of graphs whose discrete Cheeger constants are bounded by (sqrt{l}-1)^{2}/2 from below. We also show graph (or Ihara) zeta functions satisfy a certain reciprocity law.