The asymptotic spectrum of graphs, introduced by Zuiddam (Combinatorica, 2019), is the space of graph parameters that are additive under disjoint union, multiplicative under the strong product, normalized and monotone under homomorphisms between the complements. He used it to obtain a dual characterization of the Shannon capacity of graphs as the minimum of the evaluation function over the asymptotic spectrum and noted that several known upper bounds, including the Lovász number and the fractional Haemers bounds are in fact elements of the asymptotic spectrum (spectral points).

We show that every spectral point admits a probabilistic refinement and characterize the functions arising in this way. This reveals that the asymptotic spectrum can be parameterized with a convex set and the evaluation function at every graph is logarithmically convex. One consequence is that for any incomparable pair of spectral points f and g there exists a third one h and a graph G such that h(G) < min{f(G),g(G)}, thus h gives a better upper bound on the Shannon capacity of G. In addition, we show that the (logarithmic) probabilistic refinement of a spectral point on a fixed graph is the entropy function associated with a convex corner.

图的渐近谱由 Zuiddam (Combinatorica, 2019) 引入，是不相交联合下可加、强积下可乘、补间同态下归一化和单调的图参数空间。他用它来获得图的香农容量的双重表征，作为渐近谱上评估函数的最小值，并指出几个已知的上限，包括 Lovász 数和分数 Haemers 界实际上是渐近谱的元素（光谱点）。

我们表明，每个光谱点都允许概率细化，并表征以这种方式产生的函数。这表明渐近谱可以用凸集参数化，并且每个图的评估函数都是对数凸的。一个结果是，对于任何一对不可比的谱点f和g，存在第三个h和一个图G，使得h ( G ) < min{ f ( G ), g ( G )}，因此h给出了更好的上限受限于G的香农容量. 此外，我们表明固定图上谱点的（对数）概率细化是与凸角相关的熵函数。