894 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
Diameter of orientations of graphs with given order and number of blocks
A strong orientation of a graph is an assignment of a direction to each
edge such that is strongly connected. The oriented diameter of is the
smallest diameter among all strong orientations of . A block of is a
maximal connected subgraph of that has no cut vertex. A block graph is a
graph in which every block is a clique. We show that every bridgeless graph of
order containing blocks has an oriented diameter of at most . This bound is sharp for all and with .
As a corollary, we obtain a sharp upper bound on the oriented diameter in terms
of order and number of cut vertices. We also show that the oriented diameter of
a bridgeless block graph of order is bounded above by if is even and if is odd.Comment: 15 pages, 2 figure
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