We say that a graph G = (V,E) on n vertices is a β-expander for some constant β > 0 if every U ⊆ V of cardinality |U | ≤ n 2 satisfies |NG(U)| ≥ β|U | where NG(U) denotes the neighborhood of U . We explore the process of uniformly at random deleting vertices of a β-expander with probability n for some constant α > 0. Our main result implies that as n tends to infinity, the deletion process perf...

One application of graph theory is to analyze connectivity of neurons and axons in the brain. We begin with basic definitions from graph theory including the Cheeger constant, a measure of connectivity of a graph. In Section 2, we will examine expander graphs, which are very sparse yet highly connected. Surprisingly, not only do expander graphs exist, but most random graphs have the expander pr...

‎we investigate two constructions‎ - ‎the replacement and the zig-zag‎ ‎product of graphs‎ - ‎describing several fascinating connections‎ ‎with combinatorics‎, ‎via the notion of expander graph‎, ‎group‎ ‎theory‎, ‎via the notion of semidirect product and cayley graph‎, ‎and‎ ‎with markov chains‎, ‎via the lamplighter random walk‎. ‎many examples‎ ‎are provided‎.

Families of expander graphs are sparse graphs such that the number of vertices in each graph grows yet each graph remains difficult to disconnect. Expander graphs are of great importance in theoretical computer science. In this paper we study the connection between the Cheeger constant, a measure of the connectivity of the graph, and the smallest nonzero eigenvalue of the graph Laplacian. We sh...

In this lecture, we will focus on expander graphs (also called expanders), which are pseudoran-dom objects in a more restricted sense than what we saw in the last two lectures. The reader is also referred to the monograph [1] and the tutorial slides [2] for more detailed surveys of today's topics. Expander graphs are universally useful in computer science and have many applications in de-random...

In this paper we show lower bounds for a certain large class of algorithms solving the Graph Isomorphism problem, even on expander graph instances. Spielman [25] shows an algorithm for isomorphism of strongly regular expander graphs that runs in time exp{Õ(n 13 )} (this bound was recently improved to exp{Õ(n 15 )} [5]). It has since been an open question to remove the requirement that the graph...

Let G V E be an r regular expander graph Certain algorithms for nding edge disjoint paths require the edges of G to be partitioned into E E E Ek so that the graphs Gi V Ei are each expanders In this paper we give a non constructive proof of a very good split plus an algorithm which improves on that given in Broder Frieze and Upfal Existence and construction of edge disjoint paths on expander gr...

We consider the problem of testing small set expansion for general graphs. A graph G is a (k, φ)expander if every subset of volume at most k has conductance at least φ. Small set expansion has recently received significant attention due to its close connection to the unique games conjecture, the local graph partitioning algorithms and locally testable codes. We give testers with two-sided error...

We give a new lower bound on the expansion coefficient of an edge-vertex graph of a d-regular graph. As a consequence, we obtain an improvement on the lower bound on relative minimum distance of the expander codes constructed by Sipser and Spielman. We also derive some improved results on the vertex expansion of graphs that help us in improving the parameters of the expander codes of Alon, Bruc...