Combinatorial properties of the G-degree

TL;DR

This paper investigates the combinatorial properties of the G-degree in edge-colored graphs, revealing that in even dimensions, the G-degree of certain graphs is always a multiple of a factorial, affecting the structure of tensor model expansions.

Contribution

It proves that in even dimensions, the G-degree of bipartite and manifold-representing graphs is always a multiple of (d-1)! and explores implications for tensor models and 4-manifold topology.

Findings

G-degree of bipartite graphs is a multiple of (d-1)! in even dimensions

Odd powers of 1/N do not contribute in the tensor model expansion in even dimensions

In 4D, G-degree relates to regular genus and Euler characteristic of 4-manifolds

Abstract

A strong interaction is known to exist between edge-colored graphs (which encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (as a possible approach to the study of Quantum Gravity). The key tool is the {\it G-degree} of the involved graphs, which drives the {\it $1/N$ expansion} in the tensor models context. In the present paper - by making use of combinatorial properties concerning Hamiltonian decompositions of the complete graph - we prove that, in any even dimension $d\ge 4$, the G-degree of all bipartite graphs, as well as of all (bipartite or non-bipartite) graphs representing singular manifolds, is an integer multiple of $(d-1)!$. As a consequence, in even dimension, the terms of the $1/N$ expansion corresponding to odd powers of $1/N$ are null in the complex context, and do not involve colored graphs representing singular manifolds in the real context. In particular, in the 4-dimensional case, where the G-degree is shown to depend only on the regular genera with respect to an arbitrary pair of "associated" cyclic permutations, several results are obtained, relating the G-degree or the regular genus of 5-colored graphs and the Euler characteristic of the associated PL 4-manifolds.