Regular colored graphs of positive degree

TL;DR

This paper analyzes the structure and enumeration of regular colored graphs of fixed positive degree, revealing their algebraic generating functions and implications for tensor models.

Contribution

It provides the first exact and asymptotic enumeration of regular colored graphs of fixed degree and explores their generating functions and singular behavior.

Findings

Generating functions are algebraic with positive radius of convergence.

The singular behavior near the dominant singularity is characterized.

Results establish the double scaling limit of colored tensor models.

Abstract

Regular colored graphs are dual representations of pure colored D-dimensional complexes. These graphs can be classified with respect to an integer, their degree, much like maps are characterized by the genus. We analyse the structure of regular colored graphs of fixed positive degree and perform their exact and asymptotic enumeration. In particular we show that the generating function of the family of graphs of fixed degree is an algebraic series with a positive radius of convergence, independant of the degree. We describe the singular behavior of this series near its dominant singularity, and use the results to establish the double scaling limit of colored tensor models.