2-Domination and Annihilation Numbers

Using information provided by Ryan Pepper and Ermelinda DeLaVina in their paper On the 2-Domination number and Annihilation Number, I developed a new bound on the 2- domination number of trees. An original bound, γ2(G) ≤ (n+n1)/ 2 , had been shown by many other authors. Our goal was to generate a tighter bound in some cases and work towards generating a more general bound on the 2-domination number for all graphs. Throughout the span of this project I generated and proved the bound γ2(T ) ≤ 1/3 (n + 2n1 + n2). To prove this bound I first proved that the 2-domination number of a tree was less than or equal to the sum of two sub-trees formed by the deletion of an edge: γ2(T ) ≤ γ2(T1) + γ2(T2). From there, I proved our bound by showing that a minimum counter-example did not exist. A large portion of the results involves cases where a graph T is considered the minimum counter-example for the sake of contradiction. From there, I showed that if T was a counter-example, then a sub-tree T1 was also a counter-example, meaning that T would no longer be the minimum counter-example. The last portion of the results is a section comparing the two bounds on the 2-domination number with respect to the number of edges and the degrees of those edges

A characterization of trees with equal 2-domination and 2-independence numbers

A set $S$ of vertices in a graph $G$ is a $2$-dominating set if every vertex of $G$ not in $S$ is adjacent to at least two vertices in $S$, and $S$ is a $2$-independent set if every vertex in $S$ is adjacent to at most one vertex of $S$. The $2$-domination number $\gamma_2(G)$ is the minimum cardinality of a $2$-dominating set in $G$, and the $2$-independence number $\alpha_2(G)$ is the maximum cardinality of a $2$-independent set in $G$. Chellali and Meddah [{\it Trees with equal $2$-domination and $2$-independence numbers,} Discussiones Mathematicae Graph Theory 32 (2012), 263--270] provided a constructive characterization of trees with equal $2$-domination and $2$-independence numbers. Their characterization is in terms of global properties of a tree, and involves properties of minimum $2$-dominating and maximum $2$-independent sets in the tree at each stage of the construction. We provide a constructive characterization that relies only on local properties of the tree at each stage of the construction.Comment: 17 pages, 4 figure

Improved constraints on the primordial power spectrum at small scales from ultracompact minihalos

For a Gaussian spectrum of primordial density fluctuations, ultracompact minihalos (UCMHs) of dark matter are expected to be produced in much greater abundance than, e.g., primordial black holes. Forming shortly after matter-radiation equality, these objects would develop very dense and spiky dark matter profiles. In the standard scenario where dark matter consists of thermally-produced, weakly-interacting massive particles, UCMHs could thus appear as highly luminous gamma-ray sources, or leave an imprint in the cosmic microwave background by changing the reionisation history of the Universe. We derive corresponding limits on the cosmic abundance of UCMHs at different epochs, and translate them into constraints on the primordial power spectrum. We find the resulting constraints to be quite severe, especially at length scales much smaller than what can be directly probed by the cosmic microwave background or large-scale structure observations. We use our results to provide an updated compilation of the best available constraints on the power of density fluctuations on all scales, ranging from the present-day horizon to scales more than 20 orders of magnitude smaller.Comment: 7 figures, 14 pages + appendices. v2 matches version accepted for publication in PRD; updated to WMAP normalisation, updated reionisation limits, various other small changes. v3 slightly corrects the normalisation used for displaying past data in Fig 6, as well as a sign typo picked up in proof in Eq 26. All results and conclusions completely unchange