Abstract In this article, we show that the rank of 2-Steiner distance matrix a caterpillar graph having N N vertices and p p pendant veritices is 2 − 1 2N-p-1 .

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...

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...

A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I (S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S = {u, v}, then I (S) = I [u, v] is called the interval between u and v and consists of all vertices that lie on some shortest u–v path in G. T...

We prove a limit theorem for the total Steiner k-distance of a random b-ary recursive tree with weighted edges. The total Steiner k-distance is the sum of all Steiner k-distances in a tree and it generalises the Wiener index. The limit theorem is obtained by using a limit theorem in the general setting of the contraction method. In order to use the contraction method we prove a recursion formul...

The Wiener index W (G) of a connected graph G is defined as W (G) = ∑ u,v∈V (G) dG(u, v) where dG(u, v) is the distance between the vertices u and v of G. For S ⊆ V (G), the 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 k-th Steiner Wiener index SWk(G) of G is defined as SWk(G) = ∑ S⊆V (G) |S|=k d(S). We establish expressi...

This paper discusses an approach for object detection and classification. Object detection approaches find the object or objects of the real world present either in a digital image or a video, where the object can belong to any class of objects. Humans can detect the objects present in an image or video quite easily but it is not so easy to do the same by machine, for this, it is necessary to m...