Remoteness and distance, distance (signless) Laplacian eigenvalues of a graph
Let G be a connected graph of order n. The remoteness of G, denoted by ρ, is the maximum average distance from a vertex to all other vertices. Let ∂1≥⋯≥∂n$\partial_{1}\geq\cdots\geq\partial_{n}$, ∂1L≥⋯≥∂nL$\partial_{1}^{L}\geq\cdots\geq\partial_{n}^{L}$ and ∂1Q≥⋯≥∂nQ$\partial_{1} ^{Q}\geq\cdots\geq\partial_{n}^{Q}$ be the distance, distance Laplacian and distance signless Laplacian eigenvalues of G, respectively. In this paper, we give lower bounds on ρ+∂1$\rho+\partial _{1}$, ρ−∂n$\rho-\partial_{n}$, ρ+∂1L$\rho+\partial_{1}^{L}$, ∂1L−ρ$\partial_{1} ^{L}-\rho$, 2ρ+∂1Q$2\rho+\partial_{1}^{Q}$ and ∂1Q−2ρ$\partial_{1}^{Q}-2\rho$ and the corresponding extremal graphs are also characterized.
Remoteness and distance eigenvalues of a graph
Let $G$ be a connected graph of order $n$ with diameter $d$. Remoteness $\rho$ of $G$ is the maximum average distance from a vertex to all others and $\partial_1\geq\cdots\geq \partial_n$ are the distance eigenvalues of $G$. In \cite{AH}, Aouchiche and Hansen conjectured that $\rho+\partial_3>0$ when $d\geq 3$ and $\rho+\partial_{\lfloor\frac{7d}{8}\rfloor}>0.$ In this paper, we confirm these two conjectures. Furthermore, we give lower bounds on $\partial_n+\rho$ and $\partial_1-\rho$ when $G\ncong K_n$ and the extremal graphs are characterized.
Sustainable production of toxin free marine microalgae biomass as fish feed in large scale open system in the Qatari desert.
Mass cultivation of microalgae biomass for feed should be cost effective and toxin free. Evaporation loss in Qatar can be as high as 2 cm/d. Hence, production of marine microalgae biomass in Qatar would also require mitigating water loss as there was only very limited groundwater reserve. To address these issues, a combination of four growth conditions were applied to a 25,000 L raceway pond: locally isolated microalgae strain was selected which could grow in elevated salinity; strain that did not require silica and vitamins; volume of the culture would increase over time keeping denser inoculum in the beginning, and evaporation water loss would be balanced by adding seawater only. A local saline tolerant Nannochloropsis sp. was selected which did not require silica and vitamins. When the above conditions were combined in the pond, average areal biomass productivities reached 20.37 g/m(2)/d, and the culture was not contaminated by any toxic microalgae.