For weighted approximation in Lp-norm, we determine strongly asymptotic orders for the average errors of both function approximation and derivative approximation by the Bernstein operators sequence on the r-fold integrated Wiener space. AMS subject classifications: 41A28, 65D05, 41A25

Chemical compounds and drugs are often modeled as graphs where each vertex represents an atom of molecule, and covalent bounds between atoms are represented by edges between the corresponding vertices. This graph derived from a chemical compounds is often called its molecular graph, and can be different structures. In this paper, we determine the logarithm multiplicative Wiener index and recipr...

Including or excluding rare taxa in bioassessment is a controversial topic, which essentially affects the reliability and accuracy of the result. In the present paper, we hypothesize that biological indices such as Shannon-Wiener index, Simpson's index, Margalef index, evenness, BMWP (biological monitoring working party), and ASPT (Average Score Per Taxon) respond differently to rare taxa exclu...

Whereas there is an exact linear relation between the Wiener indices of kenograms and plerograms of isomeric alkanes, the respective terminal Wiener indices exhibit a completely different behavior: Correlation between terminal Wiener indices of kenograms and plerograms is absent, but other regularities can be envisaged. In this article, we analyze the basic properties of terminal Wiener indices...

The Wiener index of a graph is the sum of the distances between all pairs of vertices. In fact, many mathematicians have study the property of the sum of the distances for many years. Then later, we found that these problems have a pivotal position in studying physical properties and chemical properties of chemical molecules and many other fields. Fruitful results have been achieved on the Wien...

Let G be a graph. The distance d(u,v) between the vertices u and v of the graph G is equal to the length of a shortest path that connects u and v. The Wiener index W(G) is the sum of all distances between vertices of G, whereas the hyper-Wiener index WW(G) is defined as WW(G)=12W(G)+12@?"{"u","v"}"@?"V"("G")d (u,v)^2. In this paper the hyper-Wiener indices of the Cartesian product, composition,...

We prove a conjecture of Nadjafi-Arani et al. on the difference between the Szeged and the Wiener index of a graph (M. J. Nadjafi-Arani, H. Khodashenas, A. R. Ashrafi: Graphs whose Szeged and Wiener numbers differ by 4 and 5, Math. Comput. Modelling 55 (2012), 1644–1648). Namely, if G is a 2-connected non-complete graph on n vertices, then Sz (G) −W (G) ≥ 2n − 6. Furthermore, the equality is ob...

The Wiener index W( G) of a connected graph G is the sum of the distances d( u, v) between all pairs of vertices u and v of G. This index seems to have been introduced in [22] where it was shown that certain physical properties of various paraffin species are correlated with the Wiener index of the tree determined by the carbon atoms of the corresponding molecules. Canfield, Robinson, and Rouvr...

If G is a connected graph, then the distance between two edges is, by definition, the distance between the corresponding vertices of the line graph of G. The edge-Wiener index We of G is then equal to the sum of distances between all pairs of edges of G. We give bounds on We in terms of order and size. In particular we prove the asymptotically sharp upper bound We(G) ≤ 25 55 n5 + O(n9/2) for gr...