Topic

# Bound graph

About: Bound graph is a(n) research topic. Over the lifetime, 4216 publication(s) have been published within this topic receiving 69743 citation(s).

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##### Papers

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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.

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Abstract: We describe 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.

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1,304 citations

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Abstract: The energy, E(G), of a simple graph G is defined to be the sum of the absolute values of the eigen values of G. If G is a k-regular graph on n vertices,then E(G)⩽k+ k(n−1)(n−k) =B 2 and this bound is sharp. It is shown that for each ϵ>0, there exist infinitely many n for each of which there exists a k-regular graph G of order n with k E(G) B 2 . Two graphs with the same number of vertices are equienergetic if they have the same energy. We show that for any positive integer n⩾3, there exist two equienergetic graphs of order 4n that are not cospectral.

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834 citations

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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.

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654 citations

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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.

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Abstract: For an ordered subset W={w1,w2,…,wk} of vertices in a connected graph G and a vertex v of G, the metric representation of v with respect to W is the k-vector r(v | W)=(d(v,w 1 ) , d(v,w2),…,d(v,wk)). The set W is a resolving set for G if r(u | W)=r(v | W) implies that u=v for all pairs u,v of vertices of G. The metric dimension dim(G) of G is the minimum cardinality of a resolving set for G. Bounds on dim(G) are presented in terms of the order and the diameter of G. All connected graphs of order n having dimension 1,n−2, or n−1 are determined. A new proof for the dimension of a tree is also presented. From this result sharp bounds on the metric dimension of unicyclic graphs are established. It is shown that dim(H)⩽dim(H×K2)⩽dim(H)+1 for every connected graph H. Moreover, it is shown that for every positive real number e, there exists a connected graph G and a connected induced subgraph H of G such that dim(G)/dim(H)

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649 citations

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TL;DR: A much better bound is proved on the tree-width of planar graphs with no minor isomorphic to a g × g grid and this is the best known bound.

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Abstract: In an earlier paper, the first two authors proved that for any planar graph H , every graph with no minor isomorphic to H has bounded tree width; but the bound given there was enormous. Here we prove a much better bound. We also improve the best known bound on the tree-width of planar graphs with no minor isomorphic to a g × g grid.

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461 citations