Edge distribution in generalized graph products

Given a graph $G=(V,E)$, an integer $k$, and a function $f_G:V^k \times V^k \to {0,1}$, the $k^{th}$ graph product of $G$ w.r.t $f_G$ is the graph with vertex set $V^k$, and an edge between two vertices $x=(x_1,…,x_k)$ and $y=(y_1,…,y_k)$ iff $f_G(x,y)=1$. Graph products are a basic combinatorial object, widely studied and used in different areas such as hardness of approximation, information theory, etc. We study graph products for functions $f_G$ of the form $f_G(x,y)=1$ iff there are at least $t$ indices $i \in [k]$ s.t. $(x_i,y_i)\in E$, where $t \in [k]$ is a fixed parameter in $f_G$. This framework generalizes the well-known graph tensor-product (obtained for $t=k$) and the graph or-product (obtained for $t=1$). The property that interests us is the edge distribution in such graphs. We show that if $G$ has a spectral gap, then the number of edges connecting “large-enough” sets in $G^k$ is “well-behaved”, namely, it is close to the expected value, had the sets been random. We extend our results to bi-partite graph products as well. For a bi-partite graph $G=(X,Y,E)$, the $k^{th}$ bi-partite graph product of $G$ w.r.t $f_G$ is the bi-partite graph with vertex sets $X^k$ and $Y^k$ and edges between $x \in X^k$ and $y \in Y^k$ iff $f_G(x,y)=1$. Finally, for both types of graph products, optimality is asserted using the “Converse to the Expander Mixing Lemma” obtained by Bilu and Linial in 2006. A byproduct of our proof technique is a new explicit construction of a family of co-spectral graphs.

💡 Research Summary

The paper introduces a unified framework for graph products that interpolates between the classical tensor (or categorical) product and the OR‑product. Given a base graph (G=(V,E)), an integer (k\ge 1), and a threshold parameter (t\in{1,\dots,k}), the authors define the (k)-th generalized product (G^{k}{t}) as the graph whose vertex set is the Cartesian power (V^{k}). Two vertices (x=(x{1},\dots,x_{k})) and (y=(y_{1},\dots,y_{k})) are adjacent if and only if at least (t) coordinate pairs ((x_{i},y_{i})) belong to the edge set (E) of the original graph. The function that decides adjacency is denoted (f_{G}), and the construction reduces to the tensor product when (t=k) and to the OR‑product when (t=1).

The central question investigated is the edge distribution between large subsets of vertices in the product graph. In a random model one would expect the number of edges between two subsets (S,T\subseteq V^{k}) to be close to (p|S||T|), where (p=|E|/(|V|^{2})) is the edge density of (G). The authors prove that if the base graph (G) is an expander—more precisely, if its normalized adjacency matrix has a second eigenvalue (\lambda) that is bounded away from 1—then the same “mixing” behavior holds for (G^{k}_{t}).

To obtain this result the paper first expresses the adjacency matrix of the product as a linear combination of tensor powers of the original adjacency matrix. By exploiting the well‑known fact that eigenvalues of a tensor product are products of the eigenvalues of the factors, the authors bound the non‑trivial eigenvalues of the product matrix. The bound depends on the original spectral gap (\lambda) and on the combinatorial parameters (k) and (t); specifically, the second eigenvalue (\mu) of (G^{k}_{t}) satisfies (|\mu|\le C(k,t),\lambda^{t}) for an explicit polynomial factor (C(k,t)).

Plugging this spectral bound into the Expander Mixing Lemma yields the quantitative edge‑distribution guarantee: \