G-度的组合性质
Combinatorial properties of the G-degree
摘要：
已知边有色图（编码任意维数的PL伪流形）和随机张量模型（作为量子引力研究的一种可能方法）之间存在强相互作用。关键工具是所涉及图的{ \it G度}，它在张量模型上下文中驱动{ \it $1/N$展开}。本文利用完全图的Hamilton分解的组合性质，证明了在任意偶数维$d\ge 4$中，所有二部图以及表示奇异流形的所有（二分或非二分）图的G度都是$(d-1)!$的整数倍.因此，在偶数维中，对应于$1/N$的奇次幂的$1/N$展开的项在复上下文中为零，并且不涉及表示实数上下文中的奇异流形的有色图。特别地，在4维情形下，G度仅依赖于任意一对“关联”轮换的正则亏格，得到了几个结果，它们与5有色图的G度或正则亏格以及关联PL 4-流形的欧拉示性数有关.
英文摘要：
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.
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