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- W3130482887 abstract "For a graph H, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions W in Lp , p≥e(H) , denoted by t(H, W). One may then define corresponding functionals ‖W‖H:=|t(H,W)|1/e(H) and ‖W‖r(H):=t(H,|W|)1/e(H) , and say that H is (semi-)norming if ‖·‖H is a (semi-)norm and that H is weakly norming if ‖·‖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 ‖·‖H , we prove that ‖·‖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." @default.
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- W3130482887 date "2021-02-16" @default.
- W3130482887 modified "2023-09-27" @default.
- W3130482887 title "Two Remarks on Graph Norms" @default.
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- W3130482887 doi "https://doi.org/10.1007/s00454-021-00280-w" @default.
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