$(D+1)$-Colored Graphs - a Review of Sundry Properties

TL;DR

This paper reviews various mathematical properties of $(D+1)$-colored graphs and explores how they can be used to derive continuum metric spaces relevant to tensor model theories.

Contribution

It provides a comprehensive review of combinatorial, topological, algebraic, and metric properties of $(D+1)$-colored graphs with applications to tensor models.

Findings

Extraction of limiting continuum metric space from graphs

Calculation of critical exponents at phase transitions

Insights into properties relevant for tensor model theories

Abstract

We review the combinatorial, topological, algebraic and metric properties supported by $(D+1)$-colored graphs, with a focus on those that are pertinent to the study of tensor model theories. We show how to extract a limiting continuum metric space from this set of graphs and detail properties of this limit through the calculation of exponents at criticality.