In the complete graph on n vertices, when each edge has a weight which is an exponential random variable, Frieze proved that the minimum spanning tree has weight tending to ζ(3) = 1/13 + 1/23 + 1/33 + · · · as n → ∞. We consider spanning trees constrained to have depth bounded by k from a specified root. We prove that if k ≥ log2 log n+ω(1), where ω(1) is any function going to ∞ with n, then th...

The geodesic and geodesic interval, namely the set of all vertices lying on geodesics between a pair of vertices in a connected graph, is a part of folklore in metric graph theory. It is also known that Steiner tree of a (multi) set with k (k > 2) vertices, generalizes geodesics. In [1] the authors studied the k-Steiner intervals S(u1, u2, . . . , uk) on connected graphs (k ≥ 3) as the k-ary ge...

‎The Steiner distance of a graph‎, ‎introduced by Chartrand‎, ‎Oellermann‎, ‎Tian and Zou in 1989‎, ‎is a natural generalization of the‎ ‎concept of classical graph distance‎. ‎For a connected graph $G$ of‎ ‎order at least $2$ and $Ssubseteq V(G)$‎, ‎the Steiner‎ ‎distance $d(S)$ among the vertices of $S$ is the minimum size among‎ ‎all connected subgraphs whose vertex sets contain $S$‎. ‎Let $...

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 $2$ and $ssubseteq v(g)$, th...

Given a directed graph G = (V,E) and an integer k ≥ 1, a k-transitive-closure-spanner (k-TCspanner) of G is a directed graph H = (V,EH) that has (1) the same transitive-closure as G and (2) diameter at most k. In some applications, the shortcut paths added to the graph in order to obtain small diameter can use Steiner vertices, that is, vertices not in the original graph G. The resulting spanne...

Perfect codes and optimal anticodes in the Grassman graph Gq(n, k) are examined. It is shown that the vertices of the Grassman graph cannot be partitioned into optimal anticodes, with a possible exception when n=2k. We further examine properties of diameter perfect codes in the graph. These codes are known to be similar to Steiner systems. We discuss the connection between these systems and ``r...

Given an edge-weighted undirected graph G = (V,E, c, w), where each edge e ∈ E has a cost c(e) and a weight w(e), a set S ⊆ V of terminals and a positive constant D0, we seek a minimum cost Steiner tree where all terminals appear as leaves and its diameter is bounded by D0. Note that the diameter of a tree represents the maximum weight of path connecting two different leaves in the tree. Such p...

In this paper we give improved approximation algorithms for some network design problems. In the Bounded-Diameter or Shallow-Light k-Steiner tree problem (SLkST), we are given an undirected graph G = (V,E) with terminals T ⊆ V containing a root r ∈ T , a cost function c : E → R, a length function l : E → R, a bound L > 0 and an integer k ≥ 1. The goal is to find a minimum c-cost r-rooted Steint...

Let G be a connected graph of order n . The Steiner distance d ( S ) set vertices is the minimum size subgraph that contains all For k ≤ , -diameter sdiam maximum among sets generalises classical diameter, which coincides with 2-diameter. problem determining given order, diameter and degree was first studied by Erdös Rényi. In this paper we consider corresponding for -diameter. Δ ∈ N define e a...

the wiener index $w(g)$ of a connected graph $g$‎ ‎is defined as $w(g)=sum_{u,vin v(g)}d_g(u,v)$‎ ‎where $d_g(u,v)$ is the distance between the vertices $u$ and $v$ of‎ ‎$g$‎. ‎for $ssubseteq v(g)$‎, ‎the {it steiner distance/} $d(s)$ of‎ ‎the vertices of $s$ is the minimum size of a connected subgraph of‎ ‎$g$ whose vertex set is $s$‎. ‎the {it $k$-th steiner wiener index/}‎ ‎$sw_k(g)$ of $g$ ...