Toroidal Queens Graphs Over Finite Fields

For each positive integer n, the toroidal queens graph may be described as a graph with vertex set Zn × Zn where every vertex is adjacent to those vertices in the directions (1, 0), (0, 1), (1, 1), (1,−1) from it. We here extend this idea, examining graphs with vertex set F × F, where F is a finite field, and any four directions are used to define adjacency. The automorphism groups and isomorphism classes of such graphs are found

On Compact Symmetric Regularizations of Graphs

Let G be a finite simple graph of order n, maximum degree Δ, and minimum degree δ. A compact regularization of G is a Δ-regular graph H of which G is an induced subgraph: H is symmetric if every automorphism of G can be extended to an automorphism of H. The index |H:G| of a regularization H of G is the ratio |V(H)|/|V(G)|. Let mcr(G) denote the index of a minimum compact regularization of G and let mcsr(G) denote the index of a minimum compact symmetric regularization of G. Erdős and Kelly proved that every graph G has a compact regularization and mcr(G)≤2. Building on a result of König, Chartrand and Lesniak showed that every graph has a compact symmetric regularization and mcsr(G)≤2Δ−δ. Using a partial Cartesian product construction, we improve this to mcsr(G)≤Δ−δ+2 and give examples to show this bound cannot be reduced below Δ−δ+1

Domination of the rectangular queen's graph

The queens graph Qm×n has the squares of the m × n chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal of the board. A set D of squares of Qm×n is a dominating set for Qm×n if every square of Qm×n is either in D or adjacent to a square in D. The minimum size of a dominating set of Qm×n is the domination number, denoted by γ(Qm×n). Values of γ(Qm×n), 4 6 m 6 n 6 18, are given here, in each case with a file of minimum dominating sets (often all of them, up to symmetry) in an online appendix. In these ranges for m and n, monotonicity fails once: γ(Q8×11) = 6 > 5 = γ(Q9×11) = γ(Q10×11) = γ(Q11×11). Let g(m) [respectively g ∗ (m)] be the largest integer such that m queens suffice to dominate the (m+1)×g(m) board [respectively, to dominate the (m+1)×g ∗ (m) board with no two queens in a row]. Starting from the elementary bound g(m) 6 3m, domination when the board is far from square is investigated. It is shown (Theorem 2) that g(m) = 3m can only occur when m ≡ 0, 1, 2, 3, or 4 (mod 9), with the online appendix showing that this does occur for m 6 40, m 6= 3. Also (Theorem 4), if m ≡ 5, 6, or 7 (mod 9) then g ∗ (m) 6 3m − 2, and if m ≡ 8 (mod 9) then g ∗ (m) 6 3m − 4. It is shown that equality holds in these bounds for m 6 40. Lower bounds on γ(Qm×n) are given. In particular, if m 6 n then γ(Qm×n) > min{m, d(m + n − 2)/4e}. Two types of dominating sets (orthodox covers and centrally strong sets) are developed; each type is shown to give good upper bounds of γ(Qm×n) in several cases. Three questions are posed: whether monotonicity of γ(Qm×n) holds (other than from (m, n) = (8, 11) to (9, 11)), whether γ(Qm×n) = (m + n − 2)/4 occurs with m 6 n < 3m+ 2 (other than for (m, n) = (3, 3) and (11, 11)), and whether the lower bound given above can be improved. A set of squares is independent if no two of its squares are adjacent. The minimum size of an independent dominating set of Qm×n is the independent domination number, denoted by i(Qm×n). Values of i(Qm×n), 4 6 m 6 n 6 18, are given here, in each case with some minimum dominating sets. In these ranges for m and n, monotonicity fails twice: i(Q8×11) = 6 > 5 = i(Q9×11) = i(Q10×11) = i(Q11×11), and i(Q11×18) = 9 > 8 = i(Q12×18)

How global biodiversity hotspots may go unrecognized: Lessons from the North American Coastal Plain

© 2014 John Wiley & Sons Ltd. Biodiversity hotspots are conservation priorities. We identify the North American Coastal Plain (NACP) as a global hotspot based on the classic definition, a region with \u3e 1500 endemic plant species and \u3e 70% habitat loss. This region has been bypassed in prior designations due to misconceptions and myths about its ecology and history. These fallacies include: (1) young age of the NACP, climatic instability over time and submergence during high sea-level stands; (2) climatic and environmental homogeneity; (3) closed forest as the climax vegetation; and (4) fire regimes that are mostly anthropogenic. We show that the NACP is older and more climatically stable than usually assumed, spatially heterogeneous and extremely rich in species and endemics for its range of latitude, especially within pine savannas and other mostly herbaceous and fire-dependent communities. We suspect systematic biases and misconceptions, in addition to missing information, obscure the existence of similarly biologically significant regions world-wide. Potential solutions to this problem include (1) increased field biological surveys and taxonomic determinations, especially within grassy biomes and regions with low soil fertility, which tend to have much overlooked biodiversity; (2) more research on the climatic refugium role of hotspots, given that regions of high endemism often coincide with regions with low velocity of climate change; (3) in low-lying coastal regions, consideration of the heterogeneity in land area generated by historically fluctuating sea levels, which likely enhanced opportunities for evolution of endemic species; and (4) immediate actions to establish new protected areas and implement science-based management to restore evolutionary environmental conditions in newly recognized hotspots

Taxonomy based on science is necessary for global conservation

Peer reviewe

Improved reference genome of Aedes aegypti informs arbovirus vector control

Female Aedes aegypti mosquitoes infect more than 400 million people each year with dangerous viral pathogens including dengue, yellow fever, Zika and chikungunya. Progress in understanding the biology of mosquitoes and developing the tools to fight them has been slowed by the lack of a high-quality genome assembly. Here we combine diverse technologies to produce the markedly improved, fully re-annotated AaegL5 genome assembly, and demonstrate how it accelerates mosquito science. We anchored physical and cytogenetic maps, doubled the number of known chemosensory ionotropic receptors that guide mosquitoes to human hosts and egg-laying sites, provided further insight into the size and composition of the sex-determining M locus, and revealed copy-number variation among glutathione S-transferase genes that are important for insecticide resistance. Using high-resolution quantitative trait locus and population genomic analyses, we mapped new candidates for dengue vector competence and insecticide resistance. AaegL5 will catalyse new biological insights and intervention strategies to fight this deadly disease vector

AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 42 (2008), Pages 141–158 The automorphism group of the toroidal queen’s graph

Denote the n × n toroidal queen’s graph by Qt n. We find its automorphism group Aut(Qt n) for each positive integer n, showing that for n ≥ 6, Aut(Qt n) is generated by the translations, the group of the square, the homotheties, and (for odd n) the automorphism (x, y) ↦ → (y + x, y − x). For each n we find the automorphism classes of edges of Qt n,inparticular showing that for n&gt;1, Qt n is edge-transitive if and only if n is prime. We find the number of automorphism classes of regular solutions of the toroidal n-queens problem, generalizing work of Burger, Cockayne, and Mynhardt.