Given an ordered partition Π = {P1, P2, ..., Pt} of the vertex set V of a connected graph G = (V,E), the partition representation of a vertex v ∈ V with respect to the partition Π is the vector r(v|Π) = (d(v, P1), d(v, P2), ..., d(v, Pt)), where d(v, Pi) represents the distance between the vertex v and the set Pi. A partition Π of V is a resolving partition of G if different vertices of G have ...

The Wiener index of a connected graph G, denoted by W (G), is defined as 12 ∑ u,v∈V (G) dG(u, v). Similarly, the hyper-Wiener index of a connected graph G, denoted by WW (G), is defined as 1 2W (G) + 1 4 ∑ u,v∈V (G) dG(u, v). The vertex Padmakar-Ivan (vertex PI) index of a graph G is the sum over all edges uv of G of the number of vertices which are not equidistant from u and v. In this paper, ...

We study the kernelization complexity of structural parameterizations Vertex Cover problem. Here, goal is to find a polynomial-time preprocessing algorithm that can reduce any instance $(G,k)$ problem an equivalent one, whose size polynomial in pre-determined parameter $G$. A long line previous research deals with based on number vertex deletions needed $G$ member simple graph class $\mathcal{F...

The independent set polynomial is important in many areas of combinatorics, computer science, and statistical physics. For every integer Δ≥2, the Shearer threshold value λ⁎(Δ)=(Δ−1)Δ−1/ΔΔ. It known that for λ<−λ⁎(Δ), there are graphs G with maximum degree Δ whose polynomial, evaluated at λ, most 0. Also, no such any λ>−λ⁎(Δ). This paper motivated by computational problem approximating when λ<−λ...

A 3-path vertex cover in a graph is a vertex subset C such that every path of three vertices contains at least one vertex from C. The parameterized 3-path vertex cover problem asks whether a graph has a 3-path vertex cover of size at most k. In this paper, we give a kernel of 5k vertices and an O(1.7485)-time polynomial-space algorithm for this problem, both new results improve previous known b...

Z/nZ (n = ∏k i=1 pi ei ) are defined by extending anomalous elliptic curves over a prime filed Fp. They have n points over a ring Z/nZ and pi points over Fpi for all pi. We generalize Satoh-Araki-Smart algorithm [10], [11] and Rück algorithm [9], which solve a discrete logarithm problem over anomalous elliptic curves. We prove that a “discrete logarithm problem over super-anomalous elliptic cur...

A dendrimer is an artificially manufactured or synthesized molecule built up from branched units called monomers. In this paper, we shall give the exact formulas for the first-connectivity, second-connectivity indices and polynomials, PI and vertex-PI indices and polynomials of an infinite class of dendrimer nanostars.

The Constrained Bipartite Vertex Cover problem asks, for a bipartite graph G with partite sets A and B, and integers kA and kB , whether there is a vertex cover for G containing at most kA vertices from A and kB vertices from B. The problem has an easy kernel with 2kA · kB edges and 4kA · kB vertices, based on the fact that every vertex in A of degree more than kB has to be included in the solu...

We study the problem of decomposing the vertex set V of a graph into two parts (V1, V2) which induce subgraphs where each vertex v in V1 has degree at least a(v) and each vertex v in V2 has degree at least b(v). We investigate several polynomial cases of this NP complete problem. We give a polynomial-time algorithm for graphs with bounded treewidth which decides if a graph admits a decompositio...

The neighborhood polynomial of graph $G$ is the generating function for number vertex subsets which vertices have a common neighbor in $G$. In this paper, we investigate behavior under several operations. Specifically, provide an explicit formula obtained from given by attachment. We use result to propose recursive algorithm calculation polynomial. Finally, prove that can be found polynomial-ti...