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)
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)
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.
The main concrete result of this paper is the first explicit construction of constant degree lossless expanders. In these graphs, the expansion factor is almost as large as possible: © is an arbitrarily small constant. Such graphs are essential components in networks that can implement fast distributed, routing algorithms e.g. [PU89, ALM96, BFU99]. They are… (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 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)
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)
For any positive integers ¦ and § , let¨©¦ § denote the family of graphs on § vertices with maximum degree ¦ , and let¨©¦ § § denote the family of bipartite graphs on § vertices with § vertices in each vertex class, and with maximum degree ¦. On one hand, we note that any¨©¦ §-universal graph must have edges, respectively , such that we can efficiently find… (More)