Formal Zeta Function Expansions and the Frequency of Ramanujan Graphs

TL;DR

This paper explores the properties of the Zeta function of regular graphs and uses formal calculations to suggest that a significant proportion of large random regular graphs are Ramanujan, indicating their potential abundance.

Contribution

It introduces a formal approach linking Zeta function derivatives to graph traces and predicts the high likelihood of Ramanujan graphs among large random regular graphs.

Findings

Expected value of Zeta function derivatives has simple poles at specific points.

A majority of large random regular graphs are likely Ramanujan.

Analogue results for random covering graphs suggest similar properties.

Abstract

We show that logarithmic derivative of the Zeta function of any regular graph is given by a power series about infinity whose coefficients are given in terms of the traces of powers of the graph's Hashimoto matrix. We then consider the expected value of this power series over random, $d$-regular graph on $n$ vertices, with $d$ fixed and $n$ tending to infinity. Under rather speculative assumptions, we make a formal calculation that suggests that for fixed $d$ and $n$ large, this expected value should have simple poles of residue $-1/2$ at $\pm (d-1)^{-1/2}$. We shall explain that calculation suggests that for fixed $d$ there is an $f(d)>1/2$ such that a $d$-regular graph on $n$ vertices is Ramanujan with probability at least $f(d)$ for $n$ sufficiently large. Our formal computation has a natural analogue when we consider random covering graphs of degree $n$ over a fixed, regular "base graph." This again suggests that for $n$ large, a strict majority of random covering graphs are relatively Ramanujan. We do not regard our formal calculations as providing overwhelming evidence regarding the frequency of Ramanujan graphs. However, these calculations are quite simple, and yield intiguing suggestions which we feel merit further study.