33,533 research outputs found
Contraction semigroups on metric graphs
The main objective of the present work is to study contraction semigroups
generated by Laplace operators on metric graphs, which are not necessarily
self-adjoint. We prove criteria for such semigroups to be continuity and
positivity preserving. Also we provide a characterization of generators of
Feller semigroups on metric graphs
Metric dimension of dual polar graphs
A resolving set for a graph is a collection of vertices , chosen
so that for each vertex , the list of distances from to the members of
uniquely specifies . The metric dimension is the smallest
size of a resolving set for . We consider the metric dimension of the
dual polar graphs, and show that it is at most the rank over of
the incidence matrix of the corresponding polar space. We then compute this
rank to give an explicit upper bound on the metric dimension of dual polar
graphs.Comment: 8 page
On the strong partition dimension of graphs
We present a different way to obtain generators of metric spaces having the
property that the ``position'' of every element of the space is uniquely
determined by the distances from the elements of the generators. Specifically
we introduce a generator based on a partition of the metric space into sets of
elements. The sets of the partition will work as the new elements which will
uniquely determine the position of each single element of the space. A set
of vertices of a connected graph strongly resolves two different vertices
if either or
, where . An ordered vertex partition of
a graph is a strong resolving partition for if every two different
vertices of belonging to the same set of the partition are strongly
resolved by some set of . A strong resolving partition of minimum
cardinality is called a strong partition basis and its cardinality the strong
partition dimension. In this article we introduce the concepts of strong
resolving partition and strong partition dimension and we begin with the study
of its mathematical properties. We give some realizability results for this
parameter and we also obtain tight bounds and closed formulae for the strong
metric dimension of several graphs.Comment: 16 page
Dynamic Graphs Generators Analysis : an Illustrative Case Study
In this work, we investigate the analysis of generators for dynamic graphs,
which are defined as graphs whose topology changes over time. We introduce a
novel concept, called ''sustainability,'' to qualify the long-term evolution of
dynamic graphs. A dynamic graph is considered sustainable if its evolution does
not result in a static, empty, or periodic graph. To measure the dynamics of
the sets of vertices and edges, we propose a metric, named ''Nervousness,''
which is derived from the Jaccard distance.As an illustration of how the
analysis can be conducted, we design a parametrized generator, named D3G3
(Degree-Driven Dynamic Geometric Graphs Generator), which generates dynamic
graph instances from an initial geometric graph. The evolution of these
instances is driven by two rules that operate on the vertices based on their
degree. By varying the parameters of the generator, different properties of the
dynamic graphs can be produced.Our results show that in order to ascertain the
sustainability of the generated dynamic graphs, it is necessary to study both
the evolution of the order and the Nervousness for a given set of parameters
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