Remark Here is an interpretation of Theorem 2. Consider the following two random experiments. Experiment 1: pick a random vertex u ∈ V of the graph G, and then pick one of its d neighbors v, uniformly at random. Experiment 2: pick a random vertex u ∈ V and then pick a random vertex v ∈ V . What is the probability of picking an ordered pair (u, v) such that u ∈ B and v ∈ C? For Experiment 1, it ...

The beautiful book of Terry Tao starts with the following words: Expander graphs are a remarkable type of graph (or more precisely, a family of graphs) on finite sets of vertices that manage to simultaneously be both sparse (low-degree) and “highly connected” at the same time. They enjoy very strong mixing properties: if one starts at a fixed vertex of an (two-sided) expander graph and randomly...

An expander code is a binary linear whose parity-check matrix the bi-adjacency of bipartite graph. We provide new formula for minimum distance such codes. also proof result that $2(1-\varepsilon) \gamma n$ lower bound given by an $(m,n,d,\gamma,1-\varepsilon)$

We propose the notion of {\it resistance of a graph} as an accompanying notion of the structure entropy to measure the force of the graph to resist cascading failure of strategic virus attacks. We show that for any connected network $G$, the resistance of $G$ is $\mathcal{R}(G)=\mathcal{H}^1(G)-\mathcal{H}^2(G)$, where $\mathcal{H}^1(G)$ and $\mathcal{H}^2(G)$ are the one- and two-dimensional s...

An (n, d)-expander is a graph G = (V,E) such that for every X ⊆ V with |X| ≤ 2n− 2 we have |ΓG(X)| ≥ (d + 1)|X|. A tree T is small if it has at most n vertices and has maximum degree at most d. Friedman and Pippenger (1987) proved that any (n, d)-expander contains every small tree. However, their elegant proof does not seem to yield an efficient algorithm for obtaining the tree. In this paper, ...

T O D O :U pate gant nfo Abstract. An (n, d)-expander is a graph G = (V,E) such that for every X ⊆ V with |X| ≤ 2n − 2 we have |ΓG(X)| ≥ (d+1)|X|. A tree T is small if it has at most n vertices and has maximum degree at most d. Friedman and Pippenger (1987) proved that any (n, d)-expander contains every small tree. However, their elegant proof does not seem to yield an efficient algorithm for o...

We started out with a goal to reduce the probability of error in randomized computation. We saw that using independent runs of the randomized algorithm and then taking majority helps us amplify the success probability. However, this method uses a lot more randomness, often a costly resource, and we would like to amplify the success probability using almost the same amount of randomness as the o...

2 Powering Stage (Sketch) 2.1 Parameter Effects In this section, we will be sketchy about some details. Entering the powering stage, we have an input constraint graph denoted (G, C). G is an a (n, d, λ)-expander, with λ < d universal constants, and the constraints are over some fixed constant alphabet Σ = Σ0. Our goal is to produce a new constraint graph (G′, C ′) with a larger gap. We will den...

We prove that there exist k in and 0 < epsilon in such that every non-abelian finite simple group G, which is not a Suzuki group, has a set of k generators for which the Cayley graph Cay(G; S) is an epsilon-expander.

A work-eecient deterministic NC algorithm is presented for nding a maximum matching in a bipartite expander graph with any expansion factor > 1. This improves upon a recently presented deterministic NC maximum matching algorithm which is restricted to those bipartite expanders with large expansion factors (; > 0), and is not work-eecient 1].