The main concrete result of this paper is the first explicit construction of constant degree <i>lossless</i> expanders. In these graphs, the expansion factor is almost as large as possible: (1—ε)<i>D</i>, where <i>D</i> is the degree and ε is an arbitrarily small constant. The best previous explicit constructions gave expansion factor D/2,… (More)
We present an explicit construction of an infinite family of <i>N</i>-superconcentrators of density 44. The most economical previously known explicit graphs of this type have density around 60.
Let H be a family of graphs. A graph T is H-universal if it contains a copy of each H ∈ H as a subgraph. Let H(k, n) denote the family of graphs on n vertices with maximum degree at most k. For all positive integers k and n, we construct an H(k, n)-universal graph T with O k (n 2− 2 k log 4 k n) edges and exactly n vertices. The number of edges is almost as… (More)
Let <i>H</i> be a finite family of graphs. A graph <i>G</i> is <i>H</i>-universal if it contains a copy of each <i>H</i> ∈ <i>H</i> as a subgraph. Let <i>H</i><i>(k,n)</i> denote the family of graphs on <i>n</i> vertices with maximum degree at most <i>k.</i> For all admissible <i>k</i> and <i>n</i>, we construct an <i>H</i><i>(k, n)</i>-universal… (More)
Let 3& be the family of N-vertex graphs of maximum degree q, and with a 2-sector function j(z) < z " '. For every constant positive 6, we show via an explicit construction algorithm that there exists an 3iT3(universal graph r' of size O,(N). This construction is a significant improvement over the best previously known construction of size n(N2-").
8-ary tree. Form T' from T by first replacing each vertex v of T of height h in T with a complete 8-ary tree T,, of height 2 logs h-t logs c (or equivalently, with ch2 leaves), where c is a constant that will be described later. We now show how to interconnect the Tv's. Let 2: be any interior vertex of T, and let u be a child of v in T, and let U' E T, and… (More)
Let H be a family of graphs. We say that G is H-universal if, for each H ∈H,the graph G contains a subgraph isomorphic to H. Let H(k, n) denote the family of graphs on n vertices with maximum degree at most k. For each fixed k and each n sufficiently large, we explicitly construct an H(k, n)-universal graph Γ (k, n) with O(n 2−2/k (log n) 1+8/k) edges. This… (More)