16,247 research outputs found
On the metric dimension and fractional metric dimension for hierarchical product of graphs
A set of vertices {\em resolves} a graph if every vertex of is
uniquely determined by its vector of distances to the vertices in . The {\em
metric dimension} for , denoted by , is the minimum cardinality of
a resolving set of . In order to study the metric dimension for the
hierarchical product of two rooted graphs
and , we first introduce a new parameter, the {\em
rooted metric dimension} \rdim(G_1^{u_1}) for a rooted graph . If
is not a path with an end-vertex , we show that
\dim(G_2^{u_2}\sqcap G_1^{u_1})=|V(G_2)|\cdot\rdim(G_1^{u_1}), where
is the order of . If is a path with an end-vertex ,
we obtain some tight inequalities for .
Finally, we show that similar results hold for the fractional metric dimension.Comment: 11 page
Identifying codes in vertex-transitive graphs and strongly regular graphs
We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2ln(vertical bar V vertical bar) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order vertical bar V vertical bar(alpha) with alpha is an element of{1/4, 1/3, 2/5}These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs
Identifying codes in vertex-transitive graphs and strongly regular graphs
We consider the problem of computing identifying codes of graphs and its
fractional relaxation. The ratio between the size of optimal integer and
fractional solutions is between 1 and 2 ln(|V|)+1 where V is the set of
vertices of the graph. We focus on vertex-transitive graphs for which we can
compute the exact fractional solution. There are known examples of
vertex-transitive graphs that reach both bounds. We exhibit infinite families
of vertex-transitive graphs with integer and fractional identifying codes of
order |V|^a with a in {1/4,1/3,2/5}. These families are generalized quadrangles
(strongly regular graphs based on finite geometries). They also provide
examples for metric dimension of graphs
The Einstein Relation on Metric Measure Spaces
This note is based on F. Burghart's master thesis at Stuttgart university
from July 2018, supervised by Prof. Freiberg.
We review the Einstein relation, which connects the Hausdorff, local walk and
spectral dimensions on a space, in the abstract setting of a metric measure
space equipped with a suitable operator. This requires some twists compared to
the usual definitions from fractal geometry. The main result establishes the
invariance of the three involved notions of fractal dimension under
bi-Lipschitz continuous isomorphisms between mm-spaces and explains, more
generally, how the transport of the analytic and stochastic structure behind
the Einstein relation works. While any homeomorphism suffices for this
transport of structure, non-Lipschitz maps distort the Hausdorff and the local
walk dimension in different ways. To illustrate this, we take a look at
H\"older regular transformations and how they influence the local walk
dimension and prove some partial results concerning the Einstein relation on
graphs of fractional Brownian motions. We conclude by giving a short list of
further questions that may help building a general theory of the Einstein
relation.Comment: 28 pages, 3 figure
Fractional Local Metric Dimension of Comb Product Graphs
يعرف الرسم البياني المتصل G مع قمة الرأس (V (G ومجموعة الحافة (E (G، (حي الحل المحلي) لذرتين متجاورتين u، v بواسطة دالة الحل المحلية fi لـ G هي دالة ذات قيمة حقيقية بحيث يكون لكل رأسين متجاورين البُعد المتري المحلي الجزئي لـ الرسم البياني G يشير إلى ، وهو معرّف بواسطة وهي دالة حل محلية لـ G}. إحدى العمليات في الرسم البياني هي الرسوم البيانية لمنتج Comb. الرسوم البيانية لمنتج Comb لـ G و H يشار إليه بواسطة الهدف من هذا البحث هو تحديد البعد المتري المحلي الجزئي لـ ، وذلك لان الرسم البياني G هو رسم بياني متصل والرسم البياني H هو رسم بياني كامل نحصل من علىThe local resolving neighborhood of a pair of vertices for and is if there is a vertex in a connected graph where the distance from to is not equal to the distance from to , or defined by . A local resolving function of is a real valued function such that for and . The local fractional metric dimension of graph denoted by , defined by In this research, the author discusses about the local fractional metric dimension of comb product are two graphs, namely graph and graph , where graph is a connected graphs and graph is a complate graph and denoted by We ge
The metric dimension and metric independence of a graph
A vertex x of a graph G resolves two vertices u and v of G if the
distance from x to u does not equal the distance from x to v. A set
S of vertices of G is a resolving set for G if every two distinct vertices
of G are resolved by some vertex of S. The minimum cardinality of a
resolving set for G is called the metric dimension of G. The problem of
nding the metric dimension of a graph is formulated as an integer pro-
gramming problem. It is shown how a relaxation of this problem leads
to a linear programming problem and hence to a fractional version of
the metric dimension of a graph. The linear programming dual of this
problem is considered and the solution to the corresponding integer
programming problem is called the metric independence of the graph.
It is shown that the problem of deciding whether, for a given graph
G, the metric dimension of G equals its metric independence is NP-
complete. Trees with equal metric dimension and metric independence
are characterized. The metric independence number is established for
various classes of graphs.Preprin
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