For a graph H, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions W in \(L^p\), \(p\ge e(H)\), denoted by t(H, W). One may then define corresponding functionals \(\Vert W\Vert _{H}\,{:}{=}\,|t(H,W)|^{1/e(H)}\) and \(\Vert W\Vert _{r(H)}\,{:}{=}\,t(H,|W|)^{1/e(H)}\), 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.

对于图 H ，其在图中的同态密度自然扩展到\(L^p\)中的二元对称函数W的空间，\(p\ge e(H)\)，记为 t ( H ,  W）。然后可以定义相应的泛函\(\Vert W\Vert _{H}\,{:}{=}\,|t(H,W)|^{1/e(H)}\)和\(\ Vert W\Vert _{r(H)}\,{:}{=}\,t(H,|W|)^{1/e(H)}\)，并说H是（半）如果\(\Vert \,{\cdot }\,\Vert _{H}\)是（半）范数，则规范化，如果\(\Vert \,{\cdot }\,\Vert则H是弱规范化的_{r(H)}\)是一种规范。我们获得了两个有助于（弱）规范图理论的结果。首先，回答 Hatami 的一个问题，他估计了\(\Vert \,{\cdot }\,\Vert _{H}\)的凸模和平滑度，我们证明了\(\Vert \,{\cdot }\,\Vert _{r(H)}\)既不是一致凸的也不是一致光滑的，前提是H是弱范数。其次，我们证明每个没有孤立顶点的图H是（弱）范数当且仅当每个分量都是（弱）范数图的同构副本。这种强分解结果允许我们假设 H的连通性在研究图形规范时。特别是，我们更正了 Hatami 对上述定理的原始陈述中的一个疏忽。