The emph{Harary index} $H(G)$ of a connected graph $G$ is defined as $H(G)=sum_{u,vin V(G)}frac{1}{d_G(u,v)}$ where $d_G(u,v)$ is the distance between vertices $u$ and $v$ of $G$. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least $2$ ...

The eccentric sequence of a connected graph \(G\) is the nondecreasing eccentricities its vertices. Wiener index sum distances between all unordered pairs vertices \(G\). unique trees that minimise among with given were recently determined by present authors. In this paper we show these results hold not only for index, but large class distance-based topological indices which term Wiener-type in...

For any fixed parameter k ≥ 1, a tree k–spanner of a graph G is a spanning tree T in G such that the distance between every pair of vertices in T is at most k times their distance in G. In this paper, we generalize on this very restrictive concept, and introduce Steiner tree k–spanners: We are given an input graph consisting of terminals and Steiner vertices, and we are now looking for a tree k...

We study several network design problems with degree constraints. For the degree-constrained 2 vertex-connected subgraph problem we obtain a factor 6 violation for the degrees with 4 approximation for the cost. This improves upon the logarithmic degree violation and no cost guarantee obtained by Feder, Motwani, and Zhu (2006). Then we consider the problem of finding an arborescence with at leas...

Let G be a connected graph of order p and S a nonempty set of vertices of G. Then the Steiner distance d(S) of S is the minimum size of a connected subgraph of G whose vertex set contains S. If n is an integer, 2 ≤ n ≤ p, the Steiner n-diameter, diamn(G), of G is the maximum Steiner distance of any n-subset of vertices of G. This is a generalisation of the ordinary diameter, which is the case n...

Let G be a connected graph and S a nonempty set of vertices of G. Then the Steiner distance d,(S) of S is the smallest number of edges in a connected subgraph of G that contains S. Let k, I, s and m be nonnegative integers with m > s > 2 and k and I not both 0. Then a connected graph G is said to be k-vertex I-edge (s,m)-Steiner distance stable, if for every set S of s vertices of G with d,(S) ...

We give a simple proof of the result of Grable on the asymptotics of the number of partial Steiner systems S(t,k,m). # 2000 John Wiley & Sons, Inc.J Combin Designs 8:347±352, 2000 Keywords: partical Steiner system; matching; hypergraph 1. INTRODUCTION A partial Steiner system S…t; k;m† is a collection of k-subsets of an m-element set M such that each t-subset is contained in at most one k-subse...

Given a weighted graph G = (V,E) and a subset R of V , a Steiner tree in G is a tree which spans all vertices in R. The vertices in V \R are called Steiner vertices. A full Steiner tree is a Steiner tree in which each vertex of R is a leaf. The full Steiner tree problem is to find a full Steiner tree with minimum weight. The bottleneck full Steiner tree problem is to find a full Steiner tree wh...

Recently a new graph convexity was introduced, arising from Steiner intervals in graphs that are a natural generalization of geodesic intervals. The Steiner tree of a set W on k vertices in a connected graph G is a tree with the smallest number of edges in G that contains all vertices of W . The Steiner interval I(W ) of W consists of all vertices in G that lie on some Steiner tree with respect...

This paper studies a 4-approximation algorithm for k-prize collecting Steiner tree problems. This problem generalizes both k-minimum spanning tree problems and prize collecting Steiner tree problems. Our proposed algorithm employs two 2-approximation algorithms for k-minimum spanning tree problems and prize collecting Steiner tree problems. Also our algorithm framework can be applied to a speci...