  The aim of this paper is twofold. On the one hand, it provides a review of
the links between random tensor models, seen as quantum gravity theories, and
the PL-manifolds representation by means of edge-colored graphs
(crystallization theory). On the other hand, the core of the paper is to
establish results about the topological and geometrical properties of the
Gurau-degree (or G-degree) of the represented manifolds, in relation with the
motivations coming from physics. In fact, the G-degree appears naturally in
higher dimensional tensor models as the quantity driving their 1/N expansion,
exactly as it happens for the genus of surfaces in the two-dimensional matrix
model setting.
  In particular, the G-degree of PL-manifolds is proved to be finite-to-one in
any dimension, while in dimension 3 and 4 a series of classification theorems
are obtained for PL-manifolds represented by graphs with a fixed G-degree. All
these properties have specific relevance in the tensor models framework,
showing a direct fruitful interaction between tensor models and discrete
geometry, via crystallization theory.
