45,106 research outputs found
Identifying codes of corona product graphs
For a vertex of a graph , let be the set of with all of
its neighbors in . A set of vertices is an {\em identifying code} of
if the sets are nonempty and distinct for all vertices . If
admits an identifying code, we say that is identifiable and denote by
the minimum cardinality of an identifying code of . In this
paper, we study the identifying code of the corona product of graphs
and . We first give a necessary and sufficient condition for the
identifiable corona product , and then express in terms of and the (total) domination number of .
Finally, we compute for some special 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 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
Partitioning the vertex set of to make an efficient open domination graph
A graph is an efficient open domination graph if there exists a subset of
vertices whose open neighborhoods partition its vertex set. We characterize
those graphs for which the Cartesian product is an efficient
open domination graph when is a complete graph of order at least 3 or a
complete bipartite graph. The characterization is based on the existence of a
certain type of weak partition of . For the class of trees when is
complete of order at least 3, the characterization is constructive. In
addition, a special type of efficient open domination graph is characterized
among Cartesian products when is a 5-cycle or a 4-cycle.Comment: 16 pages, 2 figure
Loop Calculus in Statistical Physics and Information Science
Considering a discrete and finite statistical model of a general position we
introduce an exact expression for the partition function in terms of a finite
series. The leading term in the series is the Bethe-Peierls (Belief
Propagation)-BP contribution, the rest are expressed as loop-contributions on
the factor graph and calculated directly using the BP solution. The series
unveils a small parameter that often makes the BP approximation so successful.
Applications of the loop calculus in statistical physics and information
science are discussed.Comment: 4 pages, submitted to Phys.Rev.Lett. Changes: More general model,
Simpler derivatio
Homological Error Correction: Classical and Quantum Codes
We prove several theorems characterizing the existence of homological error
correction codes both classically and quantumly. Not every classical code is
homological, but we find a family of classical homological codes saturating the
Hamming bound. In the quantum case, we show that for non-orientable surfaces it
is impossible to construct homological codes based on qudits of dimension
, while for orientable surfaces with boundaries it is possible to
construct them for arbitrary dimension . We give a method to obtain planar
homological codes based on the construction of quantum codes on compact
surfaces without boundaries. We show how the original Shor's 9-qubit code can
be visualized as a homological quantum code. We study the problem of
constructing quantum codes with optimal encoding rate. In the particular case
of toric codes we construct an optimal family and give an explicit proof of its
optimality. For homological quantum codes on surfaces of arbitrary genus we
also construct a family of codes asymptotically attaining the maximum possible
encoding rate. We provide the tools of homology group theory for graphs
embedded on surfaces in a self-contained manner.Comment: Revtex4 fil
- …