Probabilistic Refinement of the Asymptotic Spectrum of Graphs

The asymptotic spectrum of graphs, introduced by Zuiddam (Combinatorica, 2019), is the space graph parameters that are additive under disjoint union, multiplicative strong product, normalized and monotone homomorphisms between complements. He used it to obtain a dual characterization Shannon capacity graphs as minimum evaluation function over noted several known upper bounds, including Lovász number fractional Haemers bounds in fact elements (spectral points). We show every spectral point admits probabilistic refinement characterize functions arising this way. This reveals can be parameterized with convex set at logarithmically convex. One consequence for any incomparable pair points f g there exists third one h G such h(G) < min{f(G),g(G)}, thus gives better bound on G. In addition, we (logarithmic) fixed entropy associated corner.