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The zero forcing polynomial of a graph
Zero forcing is an iterative graph coloring process, where given a set of
initially colored vertices, a colored vertex with a single uncolored neighbor
causes that neighbor to become colored. A zero forcing set is a set of
initially colored vertices which causes the entire graph to eventually become
colored. In this paper, we study the counting problem associated with zero
forcing. We introduce the zero forcing polynomial of a graph of order
as the polynomial , where is
the number of zero forcing sets of of size . We characterize the
extremal coefficients of , derive closed form expressions for
the zero forcing polynomials of several families of graphs, and explore various
structural properties of , including multiplicativity,
unimodality, and uniqueness.Comment: 23 page
A Comparison between the Zero Forcing Number and the Strong Metric Dimension of Graphs
The \emph{zero forcing number}, , of a graph is the minimum
cardinality of a set of black vertices (whereas vertices in are
colored white) such that is turned black after finitely many
applications of "the color-change rule": a white vertex is converted black if
it is the only white neighbor of a black vertex. The \emph{strong metric
dimension}, , of a graph is the minimum among cardinalities of all
strong resolving sets: is a \emph{strong resolving set} of
if for any , there exists an such that either
lies on an geodesic or lies on an geodesic. In this paper, we
prove that for a connected graph , where is
the cycle rank of . Further, we prove the sharp bound
when is a tree or a unicyclic graph, and we characterize trees
attaining . It is easy to see that can be
arbitrarily large for a tree ; we prove that and
show that the bound is sharp.Comment: 8 pages, 5 figure
Coordinated Beamforming with Relaxed Zero Forcing: The Sequential Orthogonal Projection Combining Method and Rate Control
In this paper, coordinated beamforming based on relaxed zero forcing (RZF)
for K transmitter-receiver pair multiple-input single-output (MISO) and
multiple-input multiple-output (MIMO) interference channels is considered. In
the RZF coordinated beamforming, conventional zero-forcing interference leakage
constraints are relaxed so that some predetermined interference leakage to
undesired receivers is allowed in order to increase the beam design space for
larger rates than those of the zero-forcing (ZF) scheme or to make beam design
feasible when ZF is impossible. In the MISO case, it is shown that the
rate-maximizing beam vector under the RZF framework for a given set of
interference leakage levels can be obtained by sequential orthogonal projection
combining (SOPC). Based on this, exact and approximate closed-form solutions
are provided in two-user and three-user cases, respectively, and an efficient
beam design algorithm for RZF coordinated beamforming is provided in general
cases. Furthermore, the rate control problem under the RZF framework is
considered. A centralized approach and a distributed heuristic approach are
proposed to control the position of the designed rate-tuple in the achievable
rate region. Finally, the RZF framework is extended to MIMO interference
channels by deriving a new lower bound on the rate of each user.Comment: Lemma 1 proof corrected; a new SOPC algorithm invented; K > N case
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