Given a connected graph G = (V, E) with non-negative edge costs, and a set of " special " nodes S ⊂ V , a subgraph of G is a Steiner tree, if it is a tree that spans (connects) all the (" special ") nodes in S. The Steiner Tree problem is to find a Steiner Tree of minimum weight (cost). Steiner Tree is an important NP-hard problem that is often encountered in practice. Examples include design o...

Given a family of graphs F , and graph G ∈ F with weights on the edges, the vertices of G are partitioned into terminals T and Steiner nodes S. The shortest paths (according to edge weights) define a metric on the set of vertices. We wish to embed the set T in a weighted graph G′ ∈ F such that the distance between any two vertices x, y ∈ T in the graph G′ is “close” to their distance in G. More...

hansen et‎. ‎al.‎, ‎using the autographix software ‎package‎, ‎conjectured that the szeged index $sz(g)$ and the‎ ‎wiener index $w(g)$ of a connected bipartite graph $g$ with $n geq ‎4$ vertices and $m geq n$ edges‎, ‎obeys the relation‎ ‎$sz(g)-w(g) geq 4n-8$‎. ‎moreover‎, ‎this bound would be the best possible‎. ‎this paper offers a proof to this conjecture‎.

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...

For a graph G and a positive integer k, the k-power of G is the graph G with V (G) as its vertex set and {(u, v)|u, v ∈ V (G), dG(u, v) ≤ k} as its edge set where dG(u, v) is the distance between u and v in graph G. The k-Steiner root problem on a graph G asks for a tree T with V (G) ⊆ V (T ) and G is the subgraph of T k induced by V (G). If such a tree T exists, we call it a k-Steiner root of ...

We combine Steiner systems with Hadamard matrices to produce a new class of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph.

let $g$ be an $(n,m)$-graph. we say that $g$ has property $(ast)$if for every pair of its adjacent vertices $x$ and $y$, thereexists a vertex $z$, such that $z$ is not adjacentto either $x$ or $y$. if the graph $g$ has property $(ast)$, thenits complement $overline g$ is connected, has diameter 2, and itswiener index is equal to $binom{n}{2}+m$, i.e., the wiener indexis insensitive of any other...

the vertex-edge wiener index of a simple connected graph g is defined as the sum of distances between vertices and edges of g. two possible distances d_1(u,e|g) and d_2(u,e|g) between a vertex u and an edge e of g were considered in the literature and according to them, the corresponding vertex-edge wiener indices w_{ve_1}(g) and w_{ve_2}(g) were introduced. in this paper, we present exact form...

Given a complete graph G = (V,E), where each vertex is labeled either terminal or Steiner, a distance function d : E → R, and a positive integer k, we study the problem of finding a Steiner tree T spanning all terminals and at most k Steiner vertices, such that the length of the longest edge is minimized. We first show that this problem is NP-hard and cannot be approximated within a factor 2− ε...