5 research outputs found
Wiener index in graphs given girth, minimum, and maximum degrees
Let be a connected graph of order . The Wiener index of is the sum of the distances between all unordered pairs of vertices of . The well-known upper bound on the Wiener index of a graph of order and minimum degree by Kouider and Winkler \cite{Kouider} was improved significantly by Alochukwu and Dankelmann \cite{Alex} for graphs containing a vertex of large degree to . In this paper, we give upper bounds on the Wiener index of in terms of order and girth , where is a function of both the minimum degree and maximum degree . Our result provides a generalisation for these previous bounds for any graph of girth . In addition, we construct graphs to show that, if for given , there exists a Moore graph of minimum degree , maximum degree and girth , then the bounds are asymptotically sharp
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