Sums over products over short paths in an expander graph

Question

Let $\Gamma=(V,E)$ be an undirected graph of degree $d$. (Say $d$ is a large constant and the number of vertices $n=|V|$ is much larger.) Let $W_0$ be the space of functions $f:V\to \mathbb{C}$ with average $0$. Assume $\Gamma$ is a strong expander graph, meaning that, for $A$ the adjacency operator $Af(w) = \sum_{\{w,v\}\in E} f(v)$ of $\Gamma$ restricted to $W_0$, all the eigenvalues of $A$ are considerably smaller than $d$. Say they are all $\leq 2 \sqrt{d}$, i.e., the graph is basically a Ramanujan graph.

Then, by definition, for all $f\in W_0$ and $\sum_{v\in V} |f(v)|^2\leq n$, $$\left|\sum_{v_1,v_2\in V: \{v_1,v_2\}\in E} f(v_1) \overline{f(v_2)}\right| \leq 2\sqrt{d} \cdot n.$$ Is it possible to give a nontrivial upper bound on $$\left|\sum_{v_1,v_2,v_3\in V: \{v_1,v_2\},\{v_2,v_3\}\in E} f(v_1) f(v_2) f(v_3)\right|?$$ Assume that $f$ is real-valued and $|f|_\infty=1$, if it helps.

(If yes: what about sums of longer products $f(v_1) f(v_2) \dotsc f(v_k)$, over $v_1,\dotsc,v_k\in V$ such that $\{v_1,v_2\},\dotsc,\{v_{k-1},v_k\}\in E$? Assume $k$ bounded.

If no: what sort of auxiliary hypothesis might help?)

Answer 1

I don't know your purpose, but here's some not-probably-sharp estimate that works for any $k$. Put $\rho=\frac{1}{d}\|A|_{({\mathbb C}1)^\perp}\|$ and $\gamma$ to be the positive root of $t^2-\rho t -\rho=0$. One has $\gamma<\sqrt{2\rho}<1$ when $\rho<\frac{1}{2}$. Then for any $f$ with $\sum f(v)=0$ and $\|f\|_\infty\le1$, one has $$\frac{1}{|V|\cdot d^{k-1}}\left|\sum_{v_1,v_2,\ldots,v_k : \{v_i,v_{i+1}\}\in E} f(v_1)\cdots f(v_k)\right| \le \gamma^k.$$

Proof. For $D:=\mathrm{diag}\,f \in B(\ell_2V)$ and $B:=\frac{1}{d}AD$, the LHS is $\frac{1}{|V|}|\langle B^{k-1}1_V,f\rangle|$. With respect to the orthogonal decomposition $\ell_2V={\mathbb C}1_V\oplus ({\mathbb C}1_V)^\perp$, one writes $B$ as an operator matrix $B=\left[\begin{smallmatrix} 0 & b \\ c & d \end{smallmatrix}\right]$, where $\| b\|\le 1$, $\|c\|\le\rho$, and $\|d\|\le\rho$. Hence for $C:=\left[\begin{smallmatrix} 0 & 1 \\ \rho & \rho \end{smallmatrix}\right] \in M_2({\mathbb R})$ with the eigenvalue $\gamma>0$ and the eigenvector $\left[\begin{smallmatrix} 1 \\ \gamma \end{smallmatrix}\right]$, one gets $$\frac{1}{|V|}|\langle B^{k-1}1_V,f\rangle| \le \left[\begin{smallmatrix} 0 & 1 \end{smallmatrix}\right] C^{k-1} \left[\begin{smallmatrix} 1 \\ 0 \end{smallmatrix}\right] \le \left[\begin{smallmatrix} 0 & 1 \end{smallmatrix}\right] C^{k-1} \left[\begin{smallmatrix} 1 \\ \gamma \end{smallmatrix}\right]=\gamma^k.$$

It's probably worth noting that the same proof shows $$\frac{1}{|V|}\sum_{v_1\in V}\left|\frac{1}{d^{k-1}}\sum_{v_2,\ldots,v_k : \{v_i,v_{i+1}\}\in E} f_1(v_1)\cdots f_k(v_k)\right|^2 \le 2\gamma^{2(k-1)}.$$