Paper Info
Title
Edge distribution in generalized graph products
Abstract
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.
Year
Venue
Field
2012
arXiv: Discrete Mathematics
Discrete mathematics,Combinatorics,Line graph,Edge-transitive graph,Graph power,Cubic graph,Cycle graph,Degree (graph theory),Distance-regular graph,Symmetric graph,Mathematics
DocType
Volume
Citations 
Journal
abs/1211.1467
0
PageRank 
References 
Authors
0.34
0
2
Name
Order
Citations
PageRank
Michael Langberg186765.83
Dan Vilenchik214313.36