Number Of Vertex Pairs Research Articles

A General Formula for Computing the Hosoya Polynomial of Capped Armchair Nanotubes

The Hosoya polynomial of a (molecular) graph G is defined as where d(G, k) is the number of vertex pairs at distance k in G. The calculation of the Hosoya polynomial of a (molecular) graph is a significant theme in Mathematical Chemistry, because from this polynomial much information about distance in the (molecular) graph, including several celebrated distance-based topological indices such as Wiener index, hyper-Wiener index, and Wiener vector, can be easily drawn. In this article, we give a general formula for computing the Hosoya polynomial of a capped armchair nanotube. We also apply this formula to derive the Hosoya polynomial of a small example of capped armchair nanotubes.

On the roots of Hosoya polynomial

Let G=(V, E) be a simple and a connected graph of diameter D. The Hosoya polynomial of G. is , where d(G, k)≥1, is the number of vertex pairs at distance k. The Wiener index of G is the first derivative of the Hosoya polynomial H(G, x) at x = 1. In this paper, we study the Hosoya polynomial and Wiener index of some family of graphs. We also discuss the roots of these family of graphs.

Approximability of Capacitated Network Design

In the capacitated survivable network design problem (Cap-SNDP), we are given an undirected multi-graph where each edge has a capacity and a cost. The goal is to find a minimum cost subset of edges that satisfies a given set of pairwise minimum-cut requirements. Unlike its classical special case of SNDP when all capacities are unit, the approximability of Cap-SNDP is not well understood; even in very restricted settings no known algorithm achieves a o(m) approximation, where m is the number of edges in the graph. In this paper, we obtain several new results and insights into the approximability of Cap-SNDP.We give an O(logn) approximation for a special case of Cap-SNDP where the global minimum cut is required to be at least R. (Note that this problem generalizes the min-cost λ-edge-connected subgraph problem, which is the special case of our problem when all capacities are unit.) Our result is based on a rounding of a natural cut-based LP relaxation strengthened with knapsack-cover (KC) inequalities. Our technique extends to give a similar approximation for a new network design problem that captures global minimum cut as a special case. We then show that as we move away from global connectivity, even for the single pair case (that is, when only one pair (s,t) has positive connectivity requirement), this strengthened LP has Ω(n) integrality gap.We also consider a variant of Cap-SNDP in which multiple copies of an edge can be bought: we give an O(logk) approximation for this case, where k is the number of vertex pairs with non-zero connectivity requirement. This improves upon the previously known O(min{k,logR max})-approximation when R max is large; here R max is the largest requirement. On the other hand, we observe that the multiple copy version of Cap-SNDP is Ω(loglogn)-hard to approximate even for the single-source version of the problem.

Hosoya polynomials of armchair open‐ended nanotubes

AbstractFor a connected graph G we denote by d(G,k) the number of vertex pairs at distance k. The Hosoya polynomial of G is H(G,x) = ∑k≥0 d(G,k)xk. In this paper, analytical formulae for calculating the polynomials of armchair open‐ended nanotubes are given. Furthermore, the Wiener index, derived from the first derivative of the Hosoya polynomial in x = 1, and the hyper‐Wiener index, from one‐half of the second derivative of the Hosoya polynomial multiplied by x in x = 1, can be calculated. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007