I would like to ask a question about Sidorenko's conjecture. Here is the background of my question:
Quasi-random graphs
A sequence of graphs $(G_n)$ is called quasi-random if it satisfies certain properties typical of random graphs. One common definition uses the spectral gap. Let $|\lambda_1| \geq |\lambda_2| \geq \dots$ be the eigenvalues of the adjacency matrix of a graph $G$. The graph $G$ is quasi-random if the ratio between the second largest eigenvalue $\lambda_2$ and the largest eigenvalue $\lambda_1$ satisfies: $$ \frac{|\lambda_2|}{|\lambda_1|} = o(1) $$ as $n \to \infty$. This means that the second largest eigenvalue becomes asymptotically small compared to the largest eigenvalue, implying that the graph behaves similarly to a random graph.
Homomorphism density
Given two graphs $H$ and $G$, the homomorphism density $t(H, G)$ is defined as the probability that a random mapping from the vertex set of $H$ to the vertex set of $G$ is a graph homomorphism (i.e., it preserves edges). Formally, for a graph $H = (V(H), E(H))$ and $G = (V(G), E(G))$, the homomorphism density is: $$ t(H, G) = \frac{|\{ \varphi : V(H) \to V(G) \,|\, \varphi \text{ is a homomorphism} \}|}{|V(G)|^{|V(H)|}}. $$
Now, in the language of graphons, we can redefine the homomorphism density. A graphon is a symmetric measurable function $W : [0, 1] \times [0, 1] \to [0, 1]$ that represents the limit of dense graphs. Using this framework, the homomorphism density $t(H, W)$ for a graphon $W$ is given by: $$ t(H, W) = \int_{[0,1]^{|V(H)|}} \prod_{(i, j) \in E(H)} W(x_i, x_j) \, dx_1 \dots dx_{|V(H)|}. $$
This expression generalizes the homomorphism density for graphs and provides a continuous model for studying extremal graph theory.
Sidorenko's conjecture
Sidorenko's conjecture asserts that for any bipartite graph $H$, and for any graph $G$, the homomorphism density satisfies: $$ t(H, G) \geq t(K_2, G)^{|E(H)|}, $$ where $K_2$ is the complete graph on two vertices, and $|E(H)|$ is the number of edges in $H$. This inequality suggests that the density of homomorphisms from bipartite graphs to any graph $G$ is at least as large as the density of edge homomorphisms raised to the power of the number of edges.
My Questions
Now, consider the following function: $$ h(H, W) = \frac{\log t(H, W)}{\log t(K_2, W)}. $$ I am studying Sidorenko's conjecture by analyzing the supremum of $h(H, W)$. Note that Sidorenko's conjecture equals to prove $h(H,G)\leq e(H)$. I have the following questions:
Q1: Is Sidorenko's conjecture equivalent to the following proposition?
For any bipartite graph $H$ and any graph $G$, $h(H, G)$ reaches its maximum if $G$ is quasi-random.
Q2: For any bipartite graph $H$, does there exist a non-sparse graphon $W$ (i.e., where $t(K_2, W) = p$ for some $p > 0$) such that $h(H, W)$ attains its maximum?
Remark: What I'm afraid of is that if $h(H, W)$ attains its maximum only when the density of $W$ tends to 0, then in this condition since $t(K_2, W)$ is also $o(1)$, $W$ being quasi-random might not imply that $h(H, W) = e(H)$.
Things seem to become more complex when dealing with sparse host graphs :(
Could this case be avodid(Q2 want to know) or be ignored whether it is sparse or not(Q1 want to know)?
