Graph norms and Sidorenko’s conjecture

By Hamed Hatami. 2008

136 citations recorded by the source corpus.

Let H and G be two finite graphs. Define hH(G) to be the number of homomorphisms from H to G. The function hH(·) extends in a natural way to a function from the set of symmetric matrices to ℝ such that for AG, the adjacency matrix of a graph G, we have hH(AG) = hH(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 hH(·)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 hH(·)1/m is a norm.We prove that hH(·)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 hH(·)1/m is a norm for certain classes of graphs, and giving some necessary conditions on the structure of H when hH(·)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 hH(·)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.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6612ee4cc1ec2627e6b8b9ddb02413182305703b

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Citations in the corpus

The History of Degenerate (Bipartite) Extremal Graph Problems (2013, methodology): “We skip the formulation of this problem, just refer to some papers of Lovász, and Hatami [140], and to the book of Lovász [177].”
Finite reflection groups and graph norms (2016, background): “In the other direction, Hatami showed that even cycles, complete bipartite graphs, and hypercubes are all weakly norming.”
Sidorenko's conjecture for blow-ups (2018, background): “Following Hatami [7], we say that a hypergraph H is weakly norming if the functional ‖ · ‖r(H) defined by ‖W‖r(H) = tH(|W”
Large Networks and Graph Limits (2012, methodology): “Is there a quantum graph g ≥ 0 which is not a square sum? Hatami and Norine [2011] constructed such a quantum graph.”
Large deviations of subgraph counts for sparse Erdős–Rényi graphs (2018, background): “ COOK AND A. DEMBO and kXkH := homH(X)1/|E|. For example, the graphs C2ℓ, ℓ≥ 1 are seminorming, with kXkC 2ℓ = kXkS 2ℓ . While a general characterization of seminorming graphs remains unknown, Hatami [Hat10] showed that complete bipartite graphs Km,n with m,nboth even are seminorming, while Conlon a”
Two Approaches to Sidorenko's Conjecture (2013, background): “Recently, Hatami [10] proved that hypercubes have Sidorenko’s property by developing a concept of norming graphs.”