On the Wiener Index of Orientations of Graphs

The Wiener index of a strong digraph $D$ is defined as the sum of the distances between all ordered pairs of vertices. This definition has been extended to digraphs that are not necessarily strong by defining the distance from a vertex $a$ to a vertex $b$ as $0$ if there is no path from $a$ to $b$ in $D$. Knor, \u{S}krekovski and Tepeh [Some remarks on Wiener index of oriented graphs. Appl.\ Math.\ Comput.\ {\bf 273}] considered orientations of graphs with maximum Wiener index. The authors conjectured that for a given tree $T$, an orientation $D$ of $T$ of maximum Wiener index always contains a vertex $v$ such that for every vertex $u$, there is either a $(u,v)$-path or a $(v,u)$-path in $D$. In this paper we disprove the conjecture. We also show that the problem of finding an orientation of maximum Wiener index of a given graph is NP-complete, thus answering a question by Knor, \u{S}krekovski and Tepeh [Orientations of graphs with maximum Wiener index. Discrete Appl.\ Math.\ 211]. We briefly discuss the corresponding problem of finding an orientation of minimum Wiener index of a given graph, and show that the special case of deciding if a given graph on $m$ edges has an orientation of Wiener index $m$ can be solved in time quadratic in $n$

Изучение кинетики сорбции отдельных компонентов композитного биосорбента

В данной статье исследуется кинетика сорбции уранил-ионов плесневыми грибами Penicillium pinophilum и Aspergillus niger. Исследования показали что степень сорбции плесневых грибов Aspergillus niger имеет на 3% большую степень сорбции урана, чем Penicillium pinophilum . Так же исследования показали, что после 12 часов сорбция заметно уменьшается и почти останавливается как у одно, так и у другого вида плесневых грибов. This article examines the kinetics of sorption of uranyl ions by fungi Penicillium pinophilum and Aspergillus niger. Studies have shown that the degree of sorption fungi Aspergillus niger has by 3% greater uranium sorption than the Penicillium pinophilum . Studies have shown that after 12 hours of sorption decreases markedly and almost stops as one or the other kind of fungi

On Complexity of Minimum Leaf Out-branching Problem

Given a digraph $D$, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in $D$ an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. Gutin, Razgon and Kim (2008) proved that MinLOB is polynomial time solvable for acyclic digraphs which are exactly the digraphs of directed path-width (DAG-width, directed tree-width, respectively) 0. We investigate how much one can extend this polynomiality result. We prove that already for digraphs of directed path-width (directed tree-width, DAG-width, respectively) 1, MinLOB is NP-hard. On the other hand, we show that for digraphs of restricted directed tree-width (directed path-width, DAG-width, respectively) and a fixed integer $k$, the problem of checking whether there is an out-branching with at most $k$ leaves is polynomial time solvable

Proximity, remoteness and maximum degree in graphs

The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are the minimum and the maximum of the average distances of the vertices of $G$, respectively. In this paper, we give upper bounds on the remoteness and proximity for graphs of given order, minimum degree and maximum degree. Our bounds are sharp apart from an additive constant.Comment: 20 page