Defensive alliances in graphs: a survey

A set $S$ of vertices of a graph $G$ is a defensive $k$-alliance in $G$ if every vertex of $S$ has at least $k$ more neighbors inside of $S$ than outside. This is primarily an expository article surveying the principal known results on defensive alliances in graph. Its seven sections are: Introduction, Computational complexity and realizability, Defensive $k$-alliance number, Boundary defensive $k$-alliances, Defensive alliances in Cartesian product graphs, Partitioning a graph into defensive $k$-alliances, and Defensive $k$-alliance free sets.Comment: 25 page

Non-comoving baryons and cold dark matter in cosmic voids

We examine the fully relativistic evolution of cosmic voids constituted by baryons and cold dark matter (CDM), represented by two non-comoving dust sources in a $\Lambda$CDM background. For this purpose, we consider numerical solutions of Einstein's field equations in a fluid-flow representation adapted to spherical symmetry and multiple components. We present a simple example that explores the frame-dependence of the local expansion and the Hubble flow for this mixture of two dusts, revealing that the relative velocity between the sources yields a significantly different evolution in comparison with that of the two sources in a common 4-velocity (which reduces to a Lemaitre-Tolman-Bondi model). In particular, significant modifications arise for the density contrast depth and void size, as well as in the amplitude of the surrounding over-densities. We show that an adequate model of a frame-dependent evolution that incorporates initial conditions from peculiar velocities and large-scale density contrast observations may contribute to understand the discrepancy between the local value of $H_0$ and that inferred from the CMB.Comment: Discussion of the evolution of baryon-CDM relative velocity added. Other minor but important corrections were incorporated. Version accepted for publication in EPJ

Effective Majorana neutrino decay

We study the decay of heavy sterile Majorana neutrinos according to the interactions obtained from an effective general theory. We describe the two and three-body decays for a wide range of neutrino masses. The results obtained and presented in this work could be useful for the study of the production and detection of this particles in a variety of high energy physics experiments and astrophysical observations. We show in different figures the dominant branching ratios and the total decay width.Comment: 22 pages, 9 figures. arXiv admin note: text overlap with arXiv:1508.0158