Let D be a digraph with vertex set V(D) .For vertex v V(D), the degree of v, denoted by d(v), is defined as the minimum value if its out-degree  and its in-degree . Now let D be a digraph with minimum degree  and edge-connectivity If  is real number, then the zeroth-order general Randic index is defined by   .  A digraph is maximally edge-connected if . In this paper we present sufficient condi...

Many graph theoretical parameters have been used to describe the vulnerability of communication networks, including toughness, binding number, rate of disruption, neighbor-connectivity, integrity, mean integrity, edgeconnectivity vector, l-connectivity and tenacity. In this paper we discuss Integrity and its properties in vulnerability calculation. The integrity of a graph G, I(G), is defined t...

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...

Vertex-connectivity and edge-connectivity represent the extent to which a graph is connected. Study of these key properties of graphs plays an important role in varieties of computer science applications. Recent years have witnessed a number of linear time 3-edge-connectivity algorithms with increasing simplicity. In contrast, the state-of-the-art algorithm for 3-vertex-connectivity due to Hopc...

We show that the graph of a simplicial polytope dimension d≥3 has no nontrivial minimum edge cut with fewer than d(d+1)/2 edges, hence is min{δ,d(d+1)/2}-edge-connected where δ denotes degree. When d=3, this implies every in plane triangulation trivial. d≥4, we construct d-polytope whose cardinality d(d+1)/2, proving aforementioned result best possible.

Recently, the authors introduced a game invariant of graphs, called connectivity. In this paper, we investigate edge version invariant, edge-connectivity, by introducing new combinatorial on graph, graph edge-cutting game. The is one generalizations classical game, named Shannon switching As an analog study have complete characterization graphs with edge-connectivity infinity in terms number ed...