Edge distribution in generalized graph products

Given a graph G and a natural number k, the k graph product of G = (V,E) is the graph with vertex set V . For every two vertices x = (x1, . . . , xk) and y = (y1, . . . , yk) in V , an edge is placed according to a predefined rule. 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 with the following “t-threshold” rule: connect every two vertices x,y ∈ V k if there are at least t indices i ∈ [k] s.t. (xi, yi) ∈ E. 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 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 bi-partite graph product of G is the bi-partite graph with vertex sets X and Y k and edges between x ∈ X and y ∈ Y k according to a predefined rule. A byproduct of our proof technique is a new explicit construction of a family of co-spectral graphs. Department of Mathematics and Computer Science, The Open University of Israel, 108 Ravutski St., Raanana 43107, Israel, email: [email protected]. Work suppoted in part by ISF grant 480/08. Dept. of Computer Science and Applied Math., The Weizmann Institute of Science, Rehovot 76100, Israel, email [email protected]. Work done in part while at The Open University of Israel.