4 research outputs found
On distance-balanced generalized Petersen graphs
A connected graph of diameter is
-distance-balanced if for every with
, where is the set of vertices of that are closer
to than to . We prove that the generalized Petersen graph is
-distance-balanced provided that is large enough
relative to . This partially solves a conjecture posed by Miklavi\v{c} and
\v{S}parl \cite{Miklavic:2018}. We also determine when
is large enough relative to
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
Non--distance-balanced generalized Petersen graphs and
A connected graph of diameter is
-distance-balanced if for every with
, where is the set of vertices of that are closer
to than to . We prove that the generalized Petersen graph
where is not -distance-balanced for any , and where is not -distance-balanced for
any . This partially solves a conjecture posed
by \v{S}. Miklavi\v{c} and P. \v{S}parl (Discrete Appl. Math. 244:143-154,
2018).Comment: 3