A Steiner triple system of order n is a collection of subsets of size three, taken from the n-element set {0, 1, ..., n−1}, such that every pair is contained in exactly one of the subsets. The subsets are called triples, and a block-intersection graph is constructed by having each triple correspond to a vertex. If two triples have a non-empty intersection, an edge is inserted between their vert...

The Harary index H can be viewed as a molecular structure descriptor composed of increments representing interactions between pairs of atoms, such that their magnitude decreases with the increasing distance between the respective two atoms. A generalization of the Harary index, denoted by Hk, is achieved by employing the Steiner-type distance between k-tuples of atoms. We show that the linear c...

Block-intersection graphs of Steiner triple systems are considered. We prove that the block-intersection graphs of non-isomorphic Steiner triple systems are themselves non-isomorphic. We also prove that each Steiner triple system of order at most 15 has a Hamilton decomposable block-intersection graph.

A family C of sets has the Helly property if any subfamily C′, whose elements are pairwise intersecting, has non-empty intersection. Suppose C is a non-empty family of subsets of a finite set V . The Helly number h(C) of C is the smallest positive integer n such that every subfamily C′ of C with |C′| ≥ n and which intersects n-wise has non-empty intersection. In this paper we consider the famil...

Given a connected graph G and a terminal set R ⊆ V (G), Steiner tree asks for a tree that includes all of R with at most r edges for some integer r ≥ 0. It is known from [ND12,Garey et. al [1]] that Steiner tree is NP-complete in general graphs. Split graph is a graph which can be partitioned into a clique and an independent set. K. White et. al [2] has established that Steiner tree in split gr...

This article introduces a proper redefinition of the concept of bottleneck Steiner distance for the Prize-Collecting Steiner Problem. This allows the application of reduction tests known to be effective on Steiner Problem in Graphs in their full power. Computational experiments attest the effectiveness of the proposed tests. keywords Steiner Problem, Network Design, Preprocessing, Combinatorial...

the reciprocal degree distance (rdd)‎, ‎defined for a connected graph $g$ as vertex-degree-weighted sum of the reciprocal distances‎, ‎that is‎, ‎$rdd(g) =sumlimits_{u,vin v(g)}frac{d_g(u)‎ + ‎d_g(v)}{d_g(u,v)}.$ the reciprocal degree distance is a weight version of the harary index‎, ‎just as the degree distance is a weight version of the wiener index‎. ‎in this paper‎, ‎we present exact formu...

If each subgraph of the decomposition π graph G has Steiner number same as then is said to be a G. The maximum cardinality among and denoted by πst(G). In this paper, we present for Complete n-Sun graph.