# Characteristic Power Series of Graph Limits

## TL;DR

This paper introduces a characteristic power series for graphons, linking spectral properties to graph quasi-randomness and offering a new perspective on spectral theory for dense graph limits.

## Contribution

It develops a characteristic power series for graphons as a limit of normalized reciprocal characteristic polynomials, providing a novel spectral characterization of graph quasi-randomness.

## Key findings

- Defines a characteristic power series for graphons.
- Connects the power series to the spectrum of the graphon.
- Provides a new spectral perspective on graph limits.

## Abstract

In this note, we show how to obtain a "characteristic power series" of graphons -- infinite limits of dense graphs -- as the limit of normalized reciprocal characteristic polynomials. This leads to a new characterization of graph quasi-randomness and another perspective on spectral theory for graphons, a complete description of the function in terms of the spectrum of the graphon as a self-adjoint kernel operator. Interestingly, while we apply a standard regularization to classical determinants, it is unclear how necessary this is.