2,835 research outputs found
Fractional Zero Forcing via Three-color Forcing Games
An -fold analogue of the positive semidefinite zero forcing process that
is carried out on the -blowup of a graph is introduced and used to define
the fractional positive semidefinite forcing number. Properties of the graph
blowup when colored with a fractional positive semidefinite forcing set are
examined and used to define a three-color forcing game that directly computes
the fractional positive semidefinite forcing number of a graph. We develop a
fractional parameter based on the standard zero forcing process and it is shown
that this parameter is exactly the skew zero forcing number with a three-color
approach. This approach and an algorithm are used to characterize graphs whose
skew zero forcing number equals zero.Comment: 24 page
Applications of analysis to the determination of the minimum number of distinct eigenvalues of a graph
We establish new bounds on the minimum number of distinct eigenvalues among
real symmetric matrices with nonzero off-diagonal pattern described by the
edges of a graph and apply these to determine the minimum number of distinct
eigenvalues of several families of graphs and small graphs
Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph
For a given graph G and an associated class of real symmetric matrices whose
off-diagonal entries are governed by the adjacencies in G, the collection of
all possible spectra for such matrices is considered. Building on the
pioneering work of Colin de Verdiere in connection with the Strong Arnold
Property, two extensions are devised that target a better understanding of all
possible spectra and their associated multiplicities. These new properties are
referred to as the Strong Spectral Property and the Strong Multiplicity
Property. Finally, these ideas are applied to the minimum number of distinct
eigenvalues associated with G, denoted by q(G). The graphs for which q(G) is at
least the number of vertices of G less one are characterized.Comment: 26 pages; corrected statement of Theorem 3.5 (a
Proof of a conjecture of Graham and Lov\'asz concerning unimodality of coefficients of the distance characteristic polynomial of a tree
We establish a conjecture of Graham and Lov\'asz that the (normalized)
coefficients of the distance characteristic polynomial of a tree are unimodal;
we also prove they are log-concave
Throttling for the game of Cops and Robbers on graphs
We consider the cop-throttling number of a graph for the game of Cops and
Robbers, which is defined to be the minimum of , where
is the number of cops and is the minimum number of
rounds needed for cops to capture the robber on over all possible
games. We provide some tools for bounding the cop-throttling number, including
showing that the positive semidefinite (PSD) throttling number, a variant of
zero forcing throttling, is an upper bound for the cop-throttling number. We
also characterize graphs having low cop-throttling number and investigate how
large the cop-throttling number can be for a given graph. We consider trees,
unicyclic graphs, incidence graphs of finite projective planes (a Meyniel
extremal family of graphs), a family of cop-win graphs with maximum capture
time, grids, and hypercubes. All the upper bounds on the cop-throttling number
we obtain for families of graphs are .Comment: 22 pages, 4 figure
Throttling positive semidefinite zero forcing propagation time on graphs
Zero forcing is a process on a graph that colors vertices blue by starting
with some of the vertices blue and applying a color change rule. Throttling
minimizes the sum of the size of the initial blue vertex set and the number of
the time steps needed to color the graph. We study throttling for positive
semidefinite zero forcing. We establish a tight lower bound on the positive
semidefinite throttling number as a function of the order, maximum degree, and
positive semidefinite zero forcing number of the graph, and determine the
positive semidefinite throttling numbers of paths, cycles, and full binary
trees. We characterize the graphs that have extreme positive semidefinite
throttling numbers.Comment: 20 pages, 7 figures, in press, Discrete Appl. Mat
Entanglement and Sources of Magnetic Anisotropy in Radical Pair-Based Avian Magnetoreceptors
One of the principal models of magnetic sensing in migratory birds rests on
the quantum spin-dynamics of transient radical pairs created photochemically in
ocular cryptochrome proteins. We consider here the role of electron spin
entanglement and coherence in determining the sensitivity of a radical
pair-based geomagnetic compass and the origins of the directional response. It
emerges that the anisotropy of radical pairs formed from spin-polarized
molecular triplets could form the basis of a more sensitive compass sensor than
one founded on the conventional hyperfine-anisotropy model. This property
offers new and more flexible opportunities for the design of biologically
inspired magnetic compass sensors
Optimizing the trade-off between number of cops and capture time in Cops and Robbers
The cop throttling number of a graph for the game of Cops and
Robbers is the minimum of , where is the number of cops and
is the minimum number of rounds needed for cops to capture the
robber on over all possible games in which both players play optimally. In
this paper, we construct a family of graphs having ,
establish a sublinear upper bound on the cop throttling number, and show that
the cop throttling number of chordal graphs is . We also introduce
the product cop throttling number as a parameter that
minimizes the person-hours used by the cops. This parameter extends the notion
of speed-up that has been studied in the context of parallel processing and
network decontamination. We establish bounds on the product cop throttling
number in terms of the cop throttling number, characterize graphs with low
product cop throttling number, and show that for a chordal graph ,
.Comment: 19 pages, 3 figure
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