326 research outputs found
On the Wiener Index of Orientations of Graphs
The Wiener index of a strong digraph 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 to a vertex as if there is no path from to in
.
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 ,
an orientation of of maximum Wiener index always contains a vertex
such that for every vertex , there is either a -path or a
-path in . 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 edges has an orientation of Wiener index
can be solved in time quadratic in
The average solution of a TSP instance in a graph
We define the average -TSP distance of a graph as the
average length of a shortest walk visiting vertices, i.e. the expected
length of the solution for a random TSP instance with uniformly random
chosen vertices. We prove relations with the average -Steiner distance and
characterize the cases where equality occurs. We also give sharp bounds for
given the order of the graph.Comment: 9 pages, 3 figure
Wiener index in graphs with given minimum degree and maximum degree
Let be a connected graph of order .The Wiener index of is
the sum of the distances between all unordered pairs of vertices of . In
this paper we show that the well-known upper bound on the Wiener index of a graph of
order and minimum degree [M. Kouider, P. Winkler, Mean distance
and minimum degree. J. Graph Theory 25 no. 1 (1997)] can be improved
significantly if the graph contains also a vertex of large degree.
Specifically, we give the asymptotically sharp bound on the Wiener
index of a graph of order , minimum degree and maximum degree
. We prove a similar result for triangle-free graphs, and we determine
a bound on the Wiener index of -free graphs of given order, minimum and
maximum degree and show that it is, in some sense, best possible
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