Roman domination number of Generalized Petersen Graphs P(n,2)

A $Roman\ domination\ function$ on a graph $G=(V, E)$ is a function $f:V(G)\rightarrow\{0,1,2\}$ satisfying the condition that every vertex $u$ with $f(u)=0$ is adjacent to at least one vertex $v$ with $f(v)=2$. The $weight$ of a Roman domination function $f$ is the value $f(V(G))=\sum_{u\in V(G)}f(u)$. The minimum weight of a Roman dominating function on a graph $G$ is called the $Roman\ domination\ number$ of $G$, denoted by $\gamma_{R}(G)$. In this paper, we study the {\it Roman domination number} of generalized Petersen graphs P(n,2) and prove that $\gamma_R(P(n,2)) = \lceil {\frac{8n}{7}}\rceil (n \geq 5)$.Comment: 9 page

The crossing number of locally twisted cubes

The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. Motivated by the recent work [Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt'o, I.: An improved upper bound on the crossing number of the hypercube. J. Graph Theory {\bf 59}, 145--161 (2008)] which solves the upper bound conjecture on the crossing number of $n$-dimensional hypercube proposed by Erd\H{o}s and Guy, we give upper and lower bounds of the crossing number of locally twisted cube, which is one of variants of hypercube.Comment: 17 pages, 12 figure

Use of surface modified porous membranes for fluid distillation

In some embodiments, the present disclosure pertains to systems and methods for distilling a fluid by exposing the fluid to a porous membrane that includes a surface capable of generating heat. In some embodiments, the heat generated at the surface propagates the distilling of the fluid by converting the fluid to a vapor that flows through the porous membrane and condenses to a distillate. In some embodiments, the surface capable of generating heat is associated with a photo-thermal composition that generates the heat at the surface by converting light energy from a light source to thermal energy. In some embodiments, the photo-thermal composition includes, without limitation, noble metals, semiconducting materials, dielectric materials, carbon-based materials, composite materials, nanocomposite materials, nanoparticles, hydrophilic materials, polymers, fibers, meshes, fiber meshes, hydrogels, hydrogel meshes, nanomaterials, and combinations thereof. Further embodiments pertain to methods of making the porous membranes of the present disclosure

\u3cem\u3eIn Situ\u3c/em\u3e Activated Co\u3csub\u3e3–x\u3c/sub\u3eNi\u3csub\u3ex\u3c/sub\u3eO\u3csub\u3e4\u3c/sub\u3e as a Highly Active and Ultrastable Electrocatalyst for Hydrogen Generation

The spinel Co3O4 has emerged as a promising alternative to noble-metal-based electrocatalysts for electrochemical water electrolysis in alkaline medium. However, pure Co3O4, despite having high activity in anodic water oxidation, remains inactive toward the hydrogen evolution reaction (HER). Here, a Ni-doped Co3O4(Co3–xNixO4) prepared by a simple method exhibits favorable HER activity and stability (\u3e300 h, whether in 1 M KOH or the realistic 30 wt % KOH solution) after in situ electrochemical activation, outperforming almost all of the oxide-based electrocatalysts. More importantly, using the combination of in situ Raman spectroscopy and multiple high-resolution electron microscopy techniques, it is identified that the surface of Co3–xNixO4 crystals is reduced into intertwined CoyNi1–yO nanoparticles with highly exposed {110} reactive planes. Density functional theory calculations further prove that the Ni-doped CoO component in CoyNi1–yO plays a major role during the alkaline HER, because the introduction of Ni atoms into Co–O octahedra can optimize the electrical conductivity and tailor the adsorption/desorption free energies of Had and OHad intermediates