Sidorenko’s conjecture states that the number of copies of a bipartite graph H in a graph G is asymptotically minimised when G is a quasirandom graph. A notorious example where this conjecture remains open is when H = K_5,5C₁₀. It was even unknown whether this graph possesses the strictly stronger, weakly norming property.

We take a step towards understanding the graph K_5,5C₁₀ by proving that it is not weakly norming. More generally, we show that ‘twisted’ blow-ups of cycles, which include K_5,5C₁₀ and C₆☐K₂, are not weakly norming. This answers two questions of Hatami. The method relies on the analysis of Hessian matrices defined by graph homomorphisms, by using the equivalence between the (weakly) norming property and convexity of graph homomorphism densities. We also prove that K_t,t minus a perfect matching, proven to be weakly norming by Lovász, is not norming for every t > 3.

Sidorenko的猜想指出，当G为准随机图时，图G中的二部图H的副本数渐近最小。当H = K _5,5 C ₁₀时，这个猜想仍然存在的一个臭名昭著的例子。甚至不知道该图是否具有严格更强，更不规范的属性。

通过证明它不是弱规范，我们朝着理解图K _5,5 C ₁₀迈出了一步。更一般地，我们表明，循环的“扭曲”吹式视窗，其中包括ķ _5,5- Ç ₁₀和C ^ ₆ ☐ ķ ₂，都没有弱拉平。这回答了Hatami的两个问题。该方法依靠图同态密度的（弱）赋范性质和凸性之间的等价关系，依靠对由图同态定义的Hessian矩阵的分析。我们还证明，K _t，t减去完美匹配（Lovász证明是弱规范性的），并不是每个t > 3都规范性的。