Abstract
For a graph H, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions W in L^p, pge e(H), denoted by t(H, W). One may then define corresponding functionals Vert WVert _{H},{:}{=},|t(H,W)|^{1/e(H)} and Vert WVert _{r(H)},{:}{=},t(H,|W|)^{1/e(H)}, and say that H is (semi-)norming if Vert ,{cdot },Vert _{H} is a (semi-)norm and that H is weakly norming if Vert ,{cdot },Vert _{r(H)} is a norm. We obtain two results that contribute to the theory of (weakly) norming graphs. Firstly, answering a question of Hatami, who estimated the modulus of convexity and smoothness of Vert ,{cdot },Vert _{H}, we prove that Vert ,{cdot },Vert _{r(H)} is neither uniformly convex nor uniformly smooth, provided that H is weakly norming. Secondly, we prove that every graph H without isolated vertices is (weakly) norming if and only if each component is an isomorphic copy of a (weakly) norming graph. This strong factorisation result allows us to assume connectivity of H when studying graph norms. In particular, we correct a negligence in the original statement of the aforementioned theorem by Hatami.
Highlights
One of the cornerstones of the theory of quasirandomness, due to Chung et al [1] and to Thomason [10], is that a graph is quasirandom if and only if it admits a randomlike count for any even cycle
A modern interpretation of this phenomenon is that the even cycle counts are essentially equivalent to the Schatten–von Neumann norms on the space of two variable symmetric functions, which are the natural limit object of large dense graphs
Graph norms have been an important concept in the theory of graph limits and received considerable attention
Summary
One of the cornerstones of the theory of quasirandomness, due to Chung et al [1] and to Thomason [10], is that a graph is quasirandom if and only if it admits a randomlike count for any even cycle. Theorem 1.1 Let H be a weakly norming graph. The normed space (Wr(H), · r(H)) is neither uniformly smooth nor uniformly convex This answers a question of Hatami, who proved that (W , · H ) is uniformly smooth and uniformly convex whenever H is semi-norming and asked for a counterpart of his theorem for weakly norming graphs. By inspecting the proof in [4], one may see that the same conclusion for weakly norming graphs H (except forests) could be obtained if · r(H) defined a uniformly convex space. Theorem 1.2 corrects a negligence that assumes connectivity of graphs without stating it, which appeared in Hatami’s work [5] and Lovász’s book [8] which study graph norms. Conlon and the third author [3, Corr. 1.3] proved that the weak factorisation result, if it exists, implies the full conjecture
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