Combinatorial properties of the G-degree
Tóm tắt
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 G-degree of the involved graphs, which drives the 1 / Nexpansion 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.
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