Connectivity augmentation is a classical problem in combinatorial optimization (see [4, 5]). Given a graph G = (V,E) and a parameter τ ∈ N, add a set of new edges E+ such that the augmented graph G′ = (V,E ∪ E+) is τ -connected (resp., τ -edge-connected). Over planar straightline graphs (PSLGs), it is NP-complete to find the minimum number of edges for τ -connectivity or τ -edge-connectivity au...

Given a graph G on n vertices with maximal degree d, form a random subgraph Gp by choosing each edge of G independently with probability p. Strengthening a classical result of Margulis we prove that if the edge connectivity k(G) satisfies k(G) À d/ log n, then the connectivity threshold in Gp is sharp. This result is asymptotically tight.

A graph G has maximal local edge-connectivity k if the maximum number of edge-disjoint paths between every pair of distinct vertices x and y is at most k. We prove Brooks-type theorems for k-connected graphs with maximal local edge-connectivity k, and for any graph with maximal local edge-connectivity 3. We also consider several related graph classes defined by constraints on connectivity. In p...

We prove an abstract version of an edge-splitting theorem for directed hypergraphs that appeared in [1], and use this result to obtain min-max theorems on hypergraph augmentation problems that are linked to orientations. These problems include (k, l)-edge-connectivity augmentation of directed hypergraphs, and (k, l)-partition-connectivity augmentation of undirected hypergraphs by uniform hypere...

The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paperweprovide anupper boundof theHarary index in terms of the vertex or edge connectivity of a graph. We characterize the unique graph with themaximumHarary index among all graphs with a given number of cut vertices or vertex connectivity or edge connectivity. In ad...

In Survivable Network problems (a.k.a. Survivable Network Design Problem – SNDP) we are given a graph with costs/weights on edges and/or nodes and prescribed connectivity requirements/demands. Among the subgraphs of G that satisfy the requirements, we seek to find one of minimum cost. Formally, the problem is defined as follows. Given a graph G = (V,E) and Q ⊆ V , the Q-connectivity λQG(uv) of ...

In this paper, the concepts of Wiener index of a vertex weighted and edge weighted graphs are discussed. Vertex weight and edge weight of a clique are introduced. Wiener index of a vertex weighted partial cube is also discussed. Also a new concept known as Connectivity index is introduced. A relation between Connectivity index and Wiener index for different graphs are discussed.

A conjecture of AutoGraphiX on the relation between the Randić index R and the algebraic connectivity a of a connected graph G is: R a ≤ ( n− 3 + 2√2 2 ) / ( 2(1− cos π n ) ) with equality if and only if G is Pn, which was proposed by Aouchiche et al. [M. Aouchiche, P. Hansen and M. Zheng, Variable neighborhood search for extremal graphs 19: further conjectures and results about the Randić inde...

For a graph G of order n, the maximum nullity of G is defined to be the largest possible nullity over all real symmetric n × n matrices A whose (i, j)th entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. Maximum nullity and the related parameter minimum rank of the same set of matrices have been studied extensively. A new parameter, maximum generic nullity, is ...