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Bound graph
About: Bound graph is a research topic. Over the lifetime, 4216 publications have been published within this topic receiving 69743 citations.
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TL;DR: An algorithm, which for fixed k ≥ 0 has running time O (| V(G) | 3 ), to solve the following problem: given a graph G and k pairs of vertices of G, decide if there are k mutually vertex-disjoint paths of G joining the pairs.
1,438 citations
TL;DR: In this article, it was shown that for any positive integer n⩾3, there exist two equienergetic graphs of order 4n that are not cospectral.
919 citations
TL;DR: Bounds on dim(G) are presented in terms of the order and the diameter of G and it is shown that dim(H)⩽dim(H×K2)⦽dim (H)+1 for every connected graph H.
821 citations
IBM1
TL;DR: In this paper, it was shown that the right-hand side is a concave function of the diagonal matrix U such that the sum of the adjacency matrix of the graph plus all the elements of the sum matrix is zero.
Abstract: Let a k-partition of a graph be a division of the vertices into k disjoint subsets containing m1 ≥ m2,..., ≥mk vertices. Let Ec be the number of edges whose two vertices belong to different subsets. Let λ1 ≥ λ2, ..., ≥ λk, be the k largest eigenvalues of a matrix, which is the sum of the adjacency matrix of the graph plus any diagonal matrix U such that the suomf all the elements of the sum matrix is zero. Then Ec ≥ 1/2Σr=1k-mrλr.
A theorem is given that shows the effect of the maximum degree of any node being limited, and it is also shown that the right-hand side is a concave function of U.C omputational studies are madoef the ratio of upper bound to lower bound for the two-partition of a number of random graphs having up to 100 nodes.
693 citations
17 May 2008
TL;DR: In this paper, a nearly-linear time algorithm was proposed to construct sparsifiers with O(n log n/e2) edges for all vectors x in RV and for any vector x in {0,1}V. This improved upon the sparsifier constructed by Spielman and Teng, which had O(m log c n) edges, and upon those of Benczur and Karger, which only satisfied (1) for x in V.
Abstract: We present a nearly-linear time algorithm that produces high-quality sparsifiers of weighted graphs. Given as input a weighted graph G=(V,E,w) and a parameter e>0, we produce a weighted subgraph H=(V,~E,~w) of G such that |~E|=O(n log n/e2) and for all vectors x in RV. (1-e) ∑uv ∈ E (x(u)-x(v))2wuv≤ ∑uv in ~E(x(u)-x(v))2~wuv ≤ (1+e)∑uv ∈ E(x(u)-x(v))2wuv. This improves upon the sparsifiers constructed by Spielman and Teng, which had O(n logc n) edges for some large constant c, and upon those of Benczur and Karger, which only satisfied (1) for x in {0,1}V. We conjecture the existence of sparsifiers with O(n) edges, noting that these would generalize the notion of expander graphs, which are constant-degree sparsifiers for the complete graph. A key ingredient in our algorithm is a subroutine of independent interest: a nearly-linear time algorithm that builds a data structure from which we can query the approximate effective resistance between any two vertices in a graph in O(log n) time.
516 citations