1,220 research outputs found
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
Distance-unbalancedness of graphs
In this paper we propose and study a new structural invariant for graphs,
called distance-unbalanced\-ness, as a measure of how much a graph is
(un)balanced in terms of distances. Explicit formulas are presented for several
classes of well-known graphs. Distance-unbalancedness of trees is also studied.
A few conjectures are stated and some open problems are proposed.Comment: 14 pages, 3 figure
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