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Connected Graph G (connected + graph_g)
Selected AbstractsKirchhoff index of linear pentagonal chainsINTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 9 2010Yan Wang Abstract The resistance distance rij between two vertices vi and vj of a connected graph G is computed as the effective resistance between nodes i and j in the corresponding network constructed from G by replacing each edge of G with a unit resistor. The Kirchhoff index Kf(G) is the sum of resistance distances between all pairs of vertices. In this article, following the method of Yang and Zhang in the proof of the Kirchhoff index of liner hexagonal chain, we obtain the closed-form formulae of the Kirchhoff index of liner pentagonal chain Pn in terms of its Laplacian spectrum. Finally, we show that the Kirchhoff index of Pn is approximately one half of its Wiener index. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010 [source] Rainbow trees in graphs and generalized connectivityNETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2010Gary Chartrand Abstract An edge-colored tree T is a rainbow tree if no two edges of T are assigned the same color. Let G be a nontrivial connected graph of order n and let k be an integer with 2 , k , n. A k -rainbow coloring of G is an edge coloring of G having the property that for every set S of k vertices of G, there exists a rainbow tree T in G such that S , V(T). The minimum number of colors needed in a k -rainbow coloring of G is the k -rainbow index of G. For every two integers k and n , 3 with 3 , k , n, the k -rainbow index of a unicyclic graph of order n is determined. For a set S of vertices in a connected graph G of order n, a collection {T1,T2,,,T,} of trees in G is said to be internally disjoint connecting S if these trees are pairwise edge-disjoint and V(Ti) , V(Tj) = S for every pair i,j of distinct integers with 1 , i,j , ,. For an integer k with 2 , k , n, the k -connectivity ,k(G) of G is the greatest positive integer , for which G contains at least , internally disjoint trees connecting S for every set S of k vertices of G. It is shown that ,k(Kn)=n,,k/2, for every pair k,n of integers with 2 , k , n. For a nontrivial connected graph G of order n and for integers k and , with 2 , k , n and 1 , , , ,k(G), the (k,,)-rainbow index rxk,,(G) of G is the minimum number of colors needed in an edge coloring of G such that G contains at least , internally disjoint rainbow trees connecting S for every set S of k vertices of G. The numbers rxk,,(Kn) are determined for all possible values k and , when n , 6. It is also shown that for , , {1, 2}, rx3,,(Kn) = 3 for all n , 6. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010 [source] Conditional diameter saturated graphsNETWORKS: AN INTERNATIONAL JOURNAL, Issue 4 2008C. Balbuena Abstract The conditional diameter D,,(G) of a connected graph G is a measure of the maximum distance between two subsets of vertices satisfying a given property ,, of interest. For any given integer k , 1, a connected graph G is said to be conditional diameter k -saturated if D,,(G) , k and there does not exist any other connected graph G, with order ,V(G,), = ,V(G),, size ,E(G,), > ,E(G),, and conditional diameter D,,(G,) , k. In this article, we obtain such conditional diameter saturated graphs for a number of properties ,,, generalizing the results obtained in (Ore, J Combin Theory 5(1968), 75,81) for the (standard) diameter D(G). © 2008 Wiley Periodicals, Inc. NETWORKS, 2008 [source] Sufficient conditions for a graph to be super restricted edge-connectedNETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2008Shiying Wang Abstract Restricted edge connectivity is a more refined network reliability index than edge connectivity. A restricted edge cut F of a connected graph G is an edge cut such that G - F has no isolated vertex. The restricted edge connectivity ,, is the minimum cardinality over all restricted edge cuts. We call G ,,-optimal if ,, = ,, where , is the minimum edge degree in G. Moreover, a ,,-optimal graph G is called a super restricted edge-connected graph if every minimum restricted edge cut separates exactly one edge. Let D and g denote the diameter and girth of G, respectively. In this paper, we first present a necessary condition for non-super restricted edge-connected graphs with minimum degree , , 3 and D , g , 2. Next, we prove that a connected graph with minimum degree , , 3 and D , g , 3 is super restricted edge-connected. Finally, we give some sufficient conditions on the conditional diameter and the girth for super restricted edge-connected graphs. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008 [source] Some results about f -critical graphsNETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2007Guizhen Liu Abstract An f -coloring of a multigraph G is a coloring of the edges of E such that each color appears at each vertex v , V at most f(v) times. The minimum number of colors needed to f -color G is called the f -chromatic index of G and is denoted by ,,f(G). Various scheduling problems on networks are reduced to finding an f -coloring of a multigraph. Any simple graph G has f -chromatic index equal to ,f(G) or ,f(G)+ 1, where ,f(G) = max v,V{, ,} and d(v) is the degree of vertex v. A connected graph G is called f -critical if ,,f(G)=,f(G)+1 and ,,f(G)=,f(G,e) < ,,f(G) for any edge e , E. Some results about f -critical graphs are given. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(3), 197,202 2007 [source] On the fault-tolerant diameter and wide diameter of ,-connected graphsNETWORKS: AN INTERNATIONAL JOURNAL, Issue 2 2005Jian-Hua Yin Abstract The fault-tolerant diameter, Dk, and wide diameter, dk, are two important parameters for measuring the reliability and efficiency of interconnection networks. It is well known that for any ,-connected graph G and any integer k, 1 , k , ,, we have Dk , dk. However, what we are interested in is how large the difference between dk and Dk can be. For any 2-connected graph G with diameter d, Flandrin and Li proved that d2 , D2 + 1 if d = 2 and d2 , (d , 1)(D2 , 1) if d , 3. In this article, we further prove that d2 , max{D2 + 1, (d , 1)(D2 , d) + 2} for d , ,(D2 , 1)/2, and d2 , max{D2 + 1,,(D2 , 1)2/4, + 2} for d , ,(D2 , 1)/2, + 1, and we also show that this upper bound can be achieved. Moreover, for any ,(, 3)-connected graph G, we prove that d, , D, + 1 if D, , 1 = 2 and d, , max{D, + 2,,(D,)2/4, + 2} if D, , 1 = 2 and D, , 1 , 3. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(2), 88,94 2005 [source] The ultracenter and central fringe of a graphNETWORKS: AN INTERNATIONAL JOURNAL, Issue 1 2001Gary Chartrand Abstract The central distance of a central vertex v in a connected graph G with rad G < diam G is the largest nonnegative integer n such that whenever x is a vertex with d(v, x) , n then x is also a central vertex. The subgraph induced by those central vertices of maximum central distance is the ultracenter of G. The subgraph induced by the central vertices having central distance 0 is the central fringe of G. For a given graph G, the smallest order of a connected graph H is determined whose ultracenter is isomorphic to G but whose center is not G. For a given graph F, we determine the smallest order of a connected graph H whose central fringe is isomorphic to G but whose center is not G. © 2001 John Wiley & Sons, Inc. [source] Minimum spanners of butterfly graphsNETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2001Shien-Ching Hwang Abstract Given a connected graph G, a spanning subgraph G, of G is called a t -spanner if every pair of two adjacent vertices in G has a distance of at most t in G,. A t -spanner of a graph G is minimum if it contains minimum number of edges among all t -spanners of G. Finding minimum spanners for general graphs is rather difficult. Most of previous results were obtained for some particular graphs, for example, butterfly graphs, cube-connected cycles, de Bruijn graphs, Kautz graphs, complete bipartite graphs, and permutation graphs. The butterfly graphs were originally introduced as the underlying graphs of FFT networks which can perform the fast Fourier transform (FFT) very efficiently. In this paper, we successfully construct most of the minimum t -spanners for the k -ary r -dimensional butterfly graphs for 2 , t , 6 and t = 8. © 2001 John Wiley & Sons, Inc. [source] Graham's pebbling conjecture on products of cyclesJOURNAL OF GRAPH THEORY, Issue 2 2003David S. Herscovici Abstract Chung defined a pebbling move on a graph G to be the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graph is the smallest number f(G) such that any distribution of f(G) pebbles on G allows one pebble to be moved to any specified, but arbitrary vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphs G and H, f(G×H), f(G)f(H). We prove Graham's conjecture when G is a cycle for a variety of graphs H, including all cycles. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 141,154, 2003 [source] | ||||||||||