Let G∗ = (V, E) be a simple graph and A any nonempty set of parameters. subset R A×V an arbitrary relation from to V. mapping ᵖ(V) written as F:A → can defined F(x) { y ∈ V/xRy} ᵖ(E) K:A K(x) {uv E/{u, v} ⊆ F(x)}. The pair (F, A) is soft over V the (K, E. Obviously (F(a), K(a)) subgraph for all A. 4-tuple G ( G∗, F, K, called G. In this paper we discuss different graphs such Complete graph, Sta...

The vertex-edge Wiener polynomials of a simple connected graph are defined based on the distances between vertices and edges of that graph. The first derivative of these polynomials at one are called the vertex-edge Wiener indices. In this paper, we express some basic properties of the first and second vertex-edge Wiener polynomials of simple connected graphs and compare the first and second ve...

To prevent an exploit, the security analyst must implement a suitable countermeasure. In this paper, we consider cost-sensitive attack graphs (CAGs) for network vulnerability analysis. In these attack graphs, a weight is assigned to each countermeasure to represent the cost of its implementation. There may be multiple countermeasures with different weights for preventing a single exploit. Also,...