图的范数与Sidorenko猜想
Graph norms and Sidorenko's conjecture
摘要：
设$H$和$G$是两个有限图.将$h_H(G)$定义为从$H$到$G$的同态数。函数$h_H(\cdot)$以自然的方式扩展到从对称矩阵集合到$\mathbb{R}$的函数，使得对于图$G$的邻接矩阵$A_G$，我们有$h_H(A_G)=h_H(G)$。令$m$为$H$的边数。很容易看出，当$H$是长度为$2n$的循环时，则$h_H(\cdot)^{1/m}$是$2n$-th Schatten-von Neumann范数。我们调查一个问题的Lovász，要求的一个表征图$H$的功能$h_H(\cdot)^{1/m}$是一个规范。证明了$h_H(\cdot)^{1/m}$是范数当且仅当Hölder型不等式对$H$成立.利用这个不等式证明了正、负结果，证明了$h_H(\cdot)^{1/m}$是某些图类的范数，并给出了当$h_H(\cdot)^{1/m}$是范数时，$H$结构的一些必要条件.作为应用，我们利用该不等式对某些含超立方体的图验证了Sidorenko的一个猜想。事实上，对于这样的图，我们可以证明比Sidorenko猜想的断言强得多的陈述。我们还调查了$h_H(\cdot)^{1/m}$规范从Banach空间理论的角度来看，确定其模的光滑性和凸性。这推广了以前已知的$2n$-th Schatten-von Neumann范数的结果。
英文摘要：
Let $H$ and $G$ be two finite graphs. Define $h_H(G)$ to be the number of homomorphisms from $H$ to $G$. The function $h_H(\cdot)$ extends in a natural way to a function from the set of symmetric matrices to $\mathbb{R}$ such that for $A_G$, the adjacency matrix of a graph $G$, we have $h_H(A_G)=h_H(G)$. Let $m$ be the number of edges of $H$. It is easy to see that when $H$ is the cycle of length $2n$, then $h_H(\cdot)^{1/m}$ is the $2n$-th Schatten-von Neumann norm. We investigate a question of Lovász that asks for a characterization of graphs $H$ for which the function $h_H(\cdot)^{1/m}$ is a norm. We prove that $h_H(\cdot)^{1/m}$ is a norm if and only if a Hölder type inequality holds for $H$. We use this inequality to prove both positive and negative results, showing that $h_H(\cdot)^{1/m}$ is a norm for certain classes of graphs, and giving some necessary conditions on the structure of $H$ when $h_H(\cdot)^{1/m}$ is a norm. As an application we use the inequality to verify a conjecture of Sidorenko for certain graphs including hypercubes. In fact for such graphs we can prove statements that are much stronger than the assertion of Sidorenko's conjecture. We also investigate the $h_H(\cdot)^{1/m}$ norms from a Banach space theoretic point of view, determining their moduli of smoothness and convexity. This generalizes the previously known result for the $2n$-th Schatten-von Neumann norms.
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