369 research outputs found
Degree Sequence Index Strategy
We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by
which to bound graph invariants by certain indices in the ordered degree
sequence. As an illustration of the DSI strategy, we show how it can be used to
give new upper and lower bounds on the -independence and the -domination
numbers. These include, among other things, a double generalization of the
annihilation number, a recently introduced upper bound on the independence
number. Next, we use the DSI strategy in conjunction with planarity, to
generalize some results of Caro and Roddity about independence number in planar
graphs. Lastly, for claw-free and -free graphs, we use DSI to
generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester
Dynamic approach to k-forcing
The k-forcing number of a graph is a generalization of the zero forcing
number. In this note, we give a greedy algorithm to approximate the k-forcing
number of a graph. Using this dynamic approach, we give corollaries which
improve upon two theorems from a recent paper of Amos, Caro, Davila and Pepper
[2], while also answering an open problem posed by Meyer [9]
Upper bounds on the k-forcing number of a graph
Given a simple undirected graph and a positive integer , the
-forcing number of , denoted , is the minimum number of vertices
that need to be initially colored so that all vertices eventually become
colored during the discrete dynamical process described by the following rule.
Starting from an initial set of colored vertices and stopping when all vertices
are colored: if a colored vertex has at most non-colored neighbors, then
each of its non-colored neighbors becomes colored. When , this is
equivalent to the zero forcing number, usually denoted with , a recently
introduced invariant that gives an upper bound on the maximum nullity of a
graph. In this paper, we give several upper bounds on the -forcing number.
Notable among these, we show that if is a graph with order and
maximum degree , then . This simplifies to, for the zero forcing number case
of , . Moreover, when and the graph is -connected, we prove that , which is an improvement when , and
specializes to, for the zero forcing number case, . These results resolve a problem posed by
Meyer about regular bipartite circulant graphs. Finally, we present a
relationship between the -forcing number and the connected -domination
number. As a corollary, we find that the sum of the zero forcing number and
connected domination number is at most the order for connected graphs.Comment: 15 pages, 0 figure
Maximum Oriented Forcing Number for Complete Graphs
The \emph{maximum oriented -forcing number} of a simple graph , written \MOF_k(G), is the maximum \emph{directed -forcing number} among all orientations of . This invariant was recently introduced by Caro, Davila and Pepper in~\cite{CaroDavilaPepper}, and in the current paper we study the special case where is the complete graph with order , denoted . While \MOF_k(G) is an invariant for the underlying simple graph , \MOF_k(K_n) can also be interpreted as an interesting property for tournaments. Our main results further focus on the case when . These include a lower bound on \MOF(K_n) of roughly , and for , a lower bound of . We also consider various lower bounds on the maximum oriented -forcing number for the closely related complete -partite graphs
Current-induced instability of domain walls in cylindrical nanowires
We study the current-driven domain wall (DW) motion in cylindrical nanowires
using micromagnetic simulations by implementing the Landau-Lifshitz-Gilbert
equation with nonlocal spin-transfer torque in a finite difference
micromagnetic package. We find that in the presence of DW Gaussian wave packets
(spin waves) will be generated when the charge current is applied to the system
suddenly. And this effect is excluded when using the local spin-transfer
torque. The existence of spin waves emission indicates that transverse domain
walls can not move arbitrarily fast in cylindrical nanowires although they are
free from the Walker limit. We establish an upper-velocity limit for the DW
motion by analyzing the stability of Gaussian wave packets using the local
spin-transfer torque. Micromagnetic simulations show that the stable region
obtained by using nonlocal spin-transfer torque is smaller than that by using
its local counterpart. This limitation is essential for multiple domain walls
since the instability of Gaussian wave packets will break the structure of
multiple domain walls.Comment: 5 pages, 6 figure
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