Let G 1⁄4 ðV ;EÞ be a k-regular graph with connectivity j and edge connectivity k. G is maximum connected if j 1⁄4 k, and G is maximum edge connected if k 1⁄4 k. Moreover, G is super-connected if it is a complete graph, or it is maximum connected and every minimum vertex cut is fxjðv; xÞ 2 Eg for some vertex v 2 V ; and G is super-edge-connected if it is maximum edge connected and every minimum...

This paper surveys the recent progress on the graph algorithms for solving network connectivity problems such as the extreme set problem, the cactus representation problem, the edge-connectivity augmentation problem and the source location problem. In particular, we show that efficient algorithms for these problems can be designed based on maximum adjacency orderings.

In this paper, we present some new lower and upper bounds for the modified Randic index in terms of maximum, minimum degree, girth, algebraic connectivity, diameter and average distance. Also we obtained relations between this index with Harmonic and Atom-bond connectivity indices. Finally, as an application we computed this index for some classes of nano-structures and linear chains.

In [1] we proposed to analyze cross-spectrum matrices obtained from electroor magneto-encephalographic (EEG/MEG) signals, to obtain estimates of the EEG/MEG sources and their coherence. In this paper we extend this method in two ways. First, by modelling such interactions as linear filters, and second, by taking the mean of the signals across different trials into account. To obtain estimates w...

The source location problem is a problem of computing a minimum cost source set in an undirected graph so that the connectivity between the source set and a vertex is at least the demand of the vertex. In this paper, the connectivity between a source set S and a vertex v is defined as the maximum number of paths between v and S no two of which have common vertex except v. We propose an O(d∗ log...

Recently, it was proved that if the diameter D of a graph G is small enough in comparison with its girth, then G is maximally connected and that a similar result also holds for digraphs. More precisely, if the diameter D of a digraph G satisfies D 5 21 1, then G has maximum connectivity ( K = 6 ) . and if D 5 21, then it attains maximum edge-connectivity ( A = 6 ) , where I is a parameter which...

A digraph G = (V, E) with diameter D is said to be s-geodetic, for 1 ≤ s ≤ D, if between any pair of (not necessarily different) vertices x, y ∈ V there is at most one x → y path of length ≤ s. Thus, any loopless digraph is at least 1-geodetic. A similar definition applies for a graph G, but in this case the concept is closely related to its girth g, for then G is s-geodetic with s = b(g − 1)/2...

Given an undirected graph G = (V,E) and subset of terminals T ⊆ V , the element-connectivity κ G (u, v) of two terminals u, v ∈ T is the maximum number of u-v paths that are pairwise disjoint in both edges and non-terminals V \ T (the paths need not be disjoint in terminals). Element-connectivity is more general than edge-connectivity and less general than vertex-connectivity. Hind and Oellerma...