4,256 research outputs found
Total Domination Dot Critical and Dot Stable Graphs.
Two vertices are said to be identifed if they are combined to form one vertex whose neighborhood is the union of their neighborhoods. A graph is total domination dot-critical if identifying any pair of adjacent vertices decreases the total domination number. On the other hand, a graph is total domination dot-stable if identifying any pair of adjacent vertices leaves the total domination number unchanged. Identifying any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most two. Among other results, we characterize total domination dot-critical trees with total domination number three and all total domination dot-stable graphs
On the diameter of dot-critical graphs
A graph G is -dot-critical (totaly -dot-critical) if is dot-critical (totaly dot-critical) and the domination number is . In the paper [T. Burtona, D. P. Sumner, Domination dot-critical graphs, Discrete Math, 306 (2006), 11-18] the following question is posed: What are the best bounds for the diameter of a -dot-critical graph and a totally -dot-critical graph with no critical vertices for ? We find the best bound for the diameter of a -dot-critical graph, where and we give a family of -dot-critical graphs (with no critical vertices) with sharp diameter for even
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices
We consider sample covariance matrices of the form , where is an matrix with independent random entries. We prove the isotropic local
Marchenko-Pastur law, i.e. we prove that the resolvent
converges to a multiple of the identity in the sense of quadratic forms. More
precisely, we establish sharp high-probability bounds on the quantity , where is the
Stieltjes transform of the Marchenko-Pastur law and . We
require the logarithms of the dimensions and to be comparable. Our
result holds down to scales and throughout the
entire spectrum away from 0. We also prove analogous results for generalized
Wigner matrices
F-Term Hybrid Inflation Followed by a Peccei-Quinn Phase Transition
We consider a cosmological set-up, based on renormalizable superpotential
terms, in which a superheavy scale F-term hybrid inflation is followed by a
Peccei-Quinn phase transition, resolving the strong CP and mu problems of the
minimal supersymmetric standard model. We show that the field which triggers
the Peccei-Quinn phase transition can remain after inflation well above the
Peccei-Quinn scale thanks to (i) its participation in the supergravity and
logarithmic corrections during the inflationary stage and (ii) the high reheat
temperature after the same period. As a consequence, its presence influences
drastically the inflationary dynamics and the universe suffers a second period
of reheating after the Peccei-Quinn phase transition. Confronting our
inflationary predictions with the current observational data, we find that, for
about the central value of the spectral index, the grand unification scale can
be identified with its supersymmetric value for the relevant coupling constant
\kappa=0.002 and, more or less, natural values, +/-(0.01-0.1), for the
remaining parameters. On the other hand, the final reheat temeperature after
the Peccei-Quinn phase transition turns out to be low enough so as the
gravitino problem is avoided.Comment: 15 pages including 8 figures, version published in Phys. Rev.
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