Garbe Frederik, Hladký Jan, Lee Joonkyung
Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67 Prague, Czechia.
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT UK.
Discrete Comput Geom. 2022;67(3):919-929. doi: 10.1007/s00454-021-00280-w. Epub 2021 Feb 16.
For a graph , its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions in , , denoted by (, ). One may then define corresponding functionals and , and say that is (semi-)norming if is a (semi-)norm and that is weakly norming if 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 , we prove that is neither uniformly convex nor uniformly smooth, provided that is weakly norming. Secondly, we prove that every graph 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 when studying graph norms. In particular, we correct a negligence in the original statement of the aforementioned theorem by Hatami.
对于一个图(G)，它在图中的同态密度自然地扩展到(x)，(y)的二元对称函数空间，记为((x,y))。然后可以定义相应的泛函(F)和(G)，并且如果(F)是一个（半）范数，则称(G)是（半）赋范的，如果(G)是一个范数，则称(G)是弱赋范的。我们得到了两个有助于（弱）赋范图理论的结果。首先，回答了哈塔米提出的一个问题，他估计了(F)的凸性和光滑性模量，我们证明如果(G)是弱赋范的，那么(F)既不是一致凸的也不是一致光滑的。其次，我们证明每个没有孤立顶点的图(G)是（弱）赋范的，当且仅当每个分量是一个（弱）赋范图的同构副本。这个强分解结果使我们在研究图范数时可以假设(G)的连通性。特别是，我们纠正了哈塔米上述定理原始陈述中的一个疏忽。
