Edge distribution in generalized graph products

Michael Langberg, Dan Vilenchik

Research output: Working paper/PreprintPreprint

Abstract

Given a graph G and a natural number k, the k th graph product of G = (V, E) is the graph with vertex set V k . For every two vertices x = (x1, . . . , xk) and y = (y1, . . . , yk) in V k
, 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 Gk 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 is the bi-partite graph with vertex sets Xk and Y k and edges between x ∈ Xk 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.
Original languageEnglish GB
Volumeabs/1211.1467
StatePublished - 2012

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NameCoRR

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