On the Steiner hyper-Wiener index of a graph

In this paper, we study the Steiner hyper-Wiener index of a graph, which is obtained from the standard hyper-Wiener index by replacing the classical graph distance with the Steiner distance. It is shown how this index is related to the Steiner Hosoya polynomial, which generalizes similar result for the standard hyper-Wiener index. Next, we show how the Steiner $3$-hyper-Wiener index of a modular graph can be expressed by using the classical graph distances. As the main result, a method for computing this index for median graphs is developed. Our method makes computation of the Steiner $3$-hyper-Wiener index much more efficient. Finally, the method is used to obtain the closed formulas for the Steiner $3$-Wiener index and the Steiner $3$-hyper-Wiener index of grid graphs

Lower bound for the cost of connecting tree with given vertex degree sequence

The optimal connecting network problem generalizes many models of structure optimization known from the literature, including communication and transport network topology design, graph cut and graph clustering, structure identification from data, etc. For the case of connecting trees with the given sequence of vertex degrees, the cost of the optimal tree is shown to be bounded from below by the solution of a semidefinite optimization program with bilinear matrix constraints, which is reduced to the solution of a series of convex programs with linear matrix inequality constraints. The proposed lower bound estimate is used to construct several heuristic algorithms and to evaluate their quality on a variety of generated and real-life data sets. Keywords: Optimal communication network, generalized Wiener index, origin-destination matrix, semidefinite programming, quadratic matrix inequality.Comment: 29 pages, 6 figures, 2 table