382 research outputs found
Entanglement-assisted zero-error source-channel coding
We study the use of quantum entanglement in the zero-error source-channel
coding problem. Here, Alice and Bob are connected by a noisy classical one-way
channel, and are given correlated inputs from a random source. Their goal is
for Bob to learn Alice's input while using the channel as little as possible.
In the zero-error regime, the optimal rates of source codes and channel codes
are given by graph parameters known as the Witsenhausen rate and Shannon
capacity, respectively. The Lov\'asz theta number, a graph parameter defined by
a semidefinite program, gives the best efficiently-computable upper bound on
the Shannon capacity and it also upper bounds its entanglement-assisted
counterpart. At the same time it was recently shown that the Shannon capacity
can be increased if Alice and Bob may use entanglement.
Here we partially extend these results to the source-coding problem and to
the more general source-channel coding problem. We prove a lower bound on the
rate of entanglement-assisted source-codes in terms Szegedy's number (a
strengthening of the theta number). This result implies that the theta number
lower bounds the entangled variant of the Witsenhausen rate. We also show that
entanglement can allow for an unbounded improvement of the asymptotic rate of
both classical source codes and classical source-channel codes. Our separation
results use low-degree polynomials due to Barrington, Beigel and Rudich,
Hadamard matrices due to Xia and Liu and a new application of remote state
preparation.Comment: Title has been changed. Previous title was 'Zero-error source-channel
coding with entanglement'. Corrected an error in Lemma 1.
A new graph based on the semi-direct product of some monoids
In this paper, firstly, we define a new graph based on the semi-direct product of a free abelian monoid of rank n by a finite cyclic monoid, and then discuss some graph properties on this new graph, namely diameter, maximum and minimum degrees, girth, degree sequence and irregularity index, domination number, chromatic number, clique number of (PM). Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in fields such as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory and physics
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
Uniqueness and minimal obstructions for tree-depth
A k-ranking of a graph G is a labeling of the vertices of G with values from
{1,...,k} such that any path joining two vertices with the same label contains
a vertex having a higher label. The tree-depth of G is the smallest value of k
for which a k-ranking of G exists. The graph G is k-critical if it has
tree-depth k and every proper minor of G has smaller tree-depth.
We establish partial results in support of two conjectures about the order
and maximum degree of k-critical graphs. As part of these results, we define a
graph G to be 1-unique if for every vertex v in G, there exists an optimal
ranking of G in which v is the unique vertex with label 1. We show that several
classes of k-critical graphs are 1-unique, and we conjecture that the property
holds for all k-critical graphs. Generalizing a previously known construction
for trees, we exhibit an inductive construction that uses 1-unique k-critical
graphs to generate large classes of critical graphs having a given tree-depth.Comment: 14 pages, 4 figure
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