10 research outputs found
Resolving sets for Johnson and Kneser graphs
A set of vertices in a graph is a {\em resolving set} for if, for
any two vertices , there exists such that the distances . In this paper, we consider the Johnson graphs and Kneser
graphs , and obtain various constructions of resolving sets for these
graphs. As well as general constructions, we show that various interesting
combinatorial objects can be used to obtain resolving sets in these graphs,
including (for Johnson graphs) projective planes and symmetric designs, as well
as (for Kneser graphs) partial geometries, Hadamard matrices, Steiner systems
and toroidal grids.Comment: 23 pages, 2 figures, 1 tabl
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
Metric Dimension of Amalgamation of Graphs
A set of vertices resolves a graph if every vertex is uniquely
determined by its vector of distances to the vertices in . The metric
dimension of is the minimum cardinality of a resolving set of .
Let be a finite collection of graphs and each
has a fixed vertex or a fixed edge called a terminal
vertex or edge, respectively. The \emph{vertex-amalgamation} of , denoted by , is formed by taking all
the 's and identifying their terminal vertices. Similarly, the
\emph{edge-amalgamation} of , denoted by
, is formed by taking all the 's and identifying
their terminal edges.
Here we study the metric dimensions of vertex-amalgamation and
edge-amalgamation for finite collection of arbitrary graphs. We give lower and
upper bounds for the dimensions, show that the bounds are tight, and construct
infinitely many graphs for each possible value between the bounds.Comment: 9 pages, 2 figures, Seventh Czech-Slovak International Symposium on
Graph Theory, Combinatorics, Algorithms and Applications (CSGT2013), revised
version 21 December 201
Partition dimension of projective planes
We determine the partition dimension of the incidence graph G(Πq) of the projective plane Πq up to a constant factor 2 as (2+o(1))log2q≤pd(G(Πq))≤(4+o(1))log2q. © 2017 Elsevier Lt
Is it possible to determine a point lying in a simplex if we know the distances from the vertices?
It is an elementary fact that if we fix an arbitrary set of d + 1 affine independent
points {p0,...,pd} in Rd, then the Euclidean distances {|x − pj |}d
j=0 determine the
point x in Rd uniquely. In this paper we investigate a similar problem in general
normed spaces which is motivated by this known fact. Namely, we characterize
those, at least d-dimensional, real normed spaces (X, · ) for which every set of
d + 1 affine independent points {p0, ...,pd} ⊂ X, the distances {x − pj}d
j=0
determine the point x lying in the simplex Conv({p0, ...,pd}) uniquely. If d = 2,
then this condition is equivalent to strict convexity, but if d > 2, then surprisingly
this holds only in inner product spaces. The core of our proof is some previously
known geometric properties of bisectors. The most important of these (Theorem 1)
is re-proven using the fundamental theorem of projective geometry
The localization number and metric dimension of graphs of diameter 2
We consider the localization number and metric dimension of certain graphs of diameter , focusing on families of Kneser graphs and graphs without 4-cycles. For the Kneser graphs with a diameter of , we find upper and lower bounds for the localization number and metric dimension, and in many cases these parameters differ only by an additive constant. Our results on the metric dimension of Kneser graphs improve on earlier ones, yielding exact values in infinitely many cases. We determine bounds on the localization number and metric dimension of Moore graphs of diameter and polarity graphs
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page