Let G be a connected graph of order n. The algebraic connectivity of G is the second smallest eigenvalue of the Laplacian matrix of G. A dominating set in G is a vertex subset S such that each vertex of G that is not in S is adjacent to a vertex in S. The least cardinality of a dominating set is the domination number. In this paper, we prove a sharp upper bound on the algebraic connectivity of ...

Using quasiconformal mappings, we prove that any Riemann surface of finite connectivity and finite genus is conformally equivalent to an intrinsic circle domain Ω in a compact Riemann surface S. This means that each connected component B of S \ Ω is either a point or a closed geometric disc with respect to the complete constant curvature conformal metric of the Riemann surface (Ω ∪ B). Moreover...

Let G be a nontrivial connected graph of order n and let k be an integer with 2 ≤ k ≤ n. For a set S of k vertices of G, let κ(S) denote the maximum number l of edge-disjoint trees T1, T2, . . . , Tl in G such that V (Ti)∩V (Tj) = S for every pair i, j of distinct integers with 1 ≤ i, j ≤ l. A collection {T1, T2, . . . , Tl} of trees in G with this property is called an internally disjoint set ...

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the first extended zeroth-order connectivity index of a graph   g is defined as 0 1/2 1 ( ) ( ) ,   v   v v g      g d      where   v   (g) is the vertex set of g, and v d is the sum of degrees of neighbors of vertex v in g. we give a sharp lower bound for the first extended zeroth-order connectivity index of trees with given numbers of vertices and pendant vertices,...

the telecom services have been recognized the world-over as an important tool for socioeconomic development for a nation and hence telecom infrastructure is treated as a crucial factor to realize the socio-economic objectives in india. accordingly, the department of telecom has been formulating developmental policies for the accelerated growth of the telecommunication services.  bsnl is a wholl...

A graph G = (V,E) is called l-mixed p-connected if G−S−L is connected for all pairs S, L with S ⊆ V , L ⊆ E, and l|S|+ |L| < p. This notion is a common generalisation of m-vertex-connectivity (l = 1, p = m) and m-edge-connectivity (l ≥ m, p = m). If p = kl then we obtain (k, l)-connectivity, introduced earlier by Kaneko and Ota, as a special case. We show that by using maximum adjacency orderin...

We introduce a model for NP-hard problems pertaining to the connectivity of graphs, and design approximation algorithms for some of the key problems in this model. One of the well-known NP-hard problems is to find a minimum-cost strongly connected spanning subgraph of a directed network. In the minimum-cost k-(S, T ) connected digraph (abbreviated, k-(S, T ) connectivity) problem we are given a...

Pacific economies are confronted with several structural constraints that limit their economic growth, most importantly their small size and remoteness from major world markets. In this paper we study the importance of transport connectivity for Pacific economies when participating in the world economy. The paper first describes the evolution of trade flows of the Pacific economies over the las...

We show that every planar, 4-connected, K2,5-minor-free graph is the square of a cycle of even length at least six.