20,439 research outputs found
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 metric dimension of affine planes, biaffine planes and generalized quadrangles
In this paper the metric dimension of (the incidence graphs of) particular
partial linear spaces is considered. We prove that the metric dimension of an
affine plane of order is and describe all resolving sets of
that size if . The metric dimension of a biaffine plane (also called
a flag-type elliptic semiplane) of order is shown to fall between
and , while for Desarguesian biaffine planes the lower bound is
improved to under , and to under certain
stronger restrictions on . We determine the metric dimension of generalized
quadrangles of order , arbitrary. We derive that the metric
dimension of generalized quadrangles of order , , is at least
, while for the classical generalized quadrangles
and it is at most
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
On the metric dimension of Grassmann graphs
The {\em metric dimension} of a graph is the least number of
vertices in a set with the property that the list of distances from any vertex
to those in the set uniquely identifies that vertex. We consider the Grassmann
graph (whose vertices are the -subspaces of , and
are adjacent if they intersect in a -subspace) for , and find a
constructive upper bound on its metric dimension. Our bound is equal to the
number of 1-dimensional subspaces of .Comment: 9 pages. Revised to correct an error in Proposition 9 of the previous
versio
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
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