12 research outputs found
Wiener Index and Remoteness in Triangulations and Quadrangulations
Let be a a connected graph. The Wiener index of a connected graph is the
sum of the distances between all unordered pairs of vertices. We provide
asymptotic formulae for the maximum Wiener index of simple triangulations and
quadrangulations with given connectivity, as the order increases, and make
conjectures for the extremal triangulations and quadrangulations based on
computational evidence. If denotes the arithmetic mean
of the distances from to all other vertices of , then the remoteness of
is defined as the largest value of over all vertices
of . We give sharp upper bounds on the remoteness of simple
triangulations and quadrangulations of given order and connectivity
Steiner Distance in Product Networks
For a connected graph of order at least and , the
\emph{Steiner distance} among the vertices of is the minimum size
among all connected subgraphs whose vertex sets contain . Let and be
two integers with . Then the \emph{Steiner -eccentricity
} of a vertex of is defined by . Furthermore, the
\emph{Steiner -diameter} of is . In this paper, we investigate the Steiner distance and Steiner
-diameter of Cartesian and lexicographical product graphs. Also, we study
the Steiner -diameter of some networks.Comment: 29 pages, 4 figure
The Maximum Wiener Index of Maximal Planar Graphs
The Wiener index of a connected graph is the sum of the distances between all
pairs of vertices in the graph. It was conjectured that the Wiener index of an
-vertex maximal planar graph is at most
. We prove this conjecture and for every
, , determine the unique -vertex maximal planar graph for
which this maximum is attained.Comment: 13 pages, 4 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