In this work extensions and variations of the notion of edge-connectivity of undirected graphs, directed graphs, and hypergraphs will be considered. We show how classical results concerning orientations and connectivity augmentations may be formulated in this more general setting.

In this paper we establish all extremal graphs with respect to augmented eccentric connectivity index among all (simple connected) graphs, among trees and among trees with perfect matching. For graphs that turn out to be extremal explicit formulas for the value of augmented eccentric connectivity index are derived.

Let ej denote the number of vertices of eccentricity j in a graph. We provide sharp lower bounds on ej in graphs of given connectivity and in chordal graphs of given connectivity.

In this article, we introduce some connectivity concepts in bipolar fuzzy graphs. Analogous to fuzzy cutvertices and fuzzy bridges in fuzzy graphs, bipolar fuzzy cutvertices and bipolar fuzzy bridges are introduced and characterized. We also propose the concepts of gain and loss for paths and pairs of vertices. Connectivity in complete bipolar fuzzy graphs is also discussed.

We determine the minimum sum–connectivity index of bicyclic graphs with n vertices and matching number m, where 2 ≤ m ≤ ⌊n2 ⌋, the minimum and the second minimum, as well as the maximum and the second maximum sum–connectivity indices of bicyclic graphs with n ≥ 5 vertices. The extremal graphs are characterized. MSC 2000: 05C90; 05C35; 05C07

We consider the problems of finding k-connected spanning subgraphs with minimum average weight. We show that the problems are NP-hard for k > 1. Approximation algorithms are given for four versions of the minimum average edge weight problem: 1. 3-approximation for k-edge-connectivity, 2. O(logk) approximation for k-node-connectivity 3. 2+ approximation for k-node-connectivity in Euclidian graph...

This paper surveys extremal properties of general sum-connectivity index χα(G) in several classes of connected graphs of given order for some values of the parameter α: a) trees; b) connected unicyclic or bicyclic graphs; c) graphs of given connectivity. Work presented as invited lecture at CAIM 2014, September 19-22, “Vasile Alecsandri” University of Bacău, Romania.

The generalized k-connectivity κk(G) of a graph G was introduced by Hager before 1985. As its a natural counterpart, we introduced the concept of generalized edge-connectivity λk(G), recently. In this paper we summarize the known results on the generalized connectivity and generalized edge-connectivity. After an introductory section, the paper is then divided into nine sections: the generalized...

We survey some recent results on 2-edge and 2-vertex connectivity problems in directed graphs. Despite being complete analogs of the corresponding notions on undirected graphs, in digraphs 2vertex and 2-edge connectivity have a much richer and more complicated structure. It is thus not surprising that 2-connectivity problems on directed graphs appear to be more difficult than on undirected grap...