Degree Sequences and Long Cycles in Graphs

Let $G$ be a graph on $n$ vertices with degree sequence $\delta=d_1\le d_2\le...\le d_n$. Let $c$ be the circumference - the order of a longest cycle and $p$ the order of a longest path in $G$. In 1952, Dirac proved: (i) every graph with $2d_1\ge n$ is hamiltonian; (ii) in every 2-connected graph, $c\ge \min\{p,2d_1\}$. Recently, the bounds $2d_1\ge n$ and $c\ge \min\{p,2d_1\}$ in (i) and (ii) are improved to $2d_\delta\ge n$ and $c\ge \min\{p,2d_\delta\}$, respectively, by Koulakzian, Mosesyan and Nikoghosyan. In this paper we present two new sharp bounds $d_\delta+d_{\delta+1}\ge n$ and $\min\{2d_{\delta+1},d_\delta+d_{\delta+2}\}\ge n$ instead of $2d_\delta\ge n$, as well as two new sharp bounds $c\ge \min\{p,d_\delta+d_{\delta+1}\}$ and $c\ge \min\{p,2d_{\delta+1},d_\delta+d_{\delta+2}\}$ instead of $c\ge \min\{p,2d_\delta\}$.Comment: 9 page

Stability of Ge-related point defects and complexes in Ge-doped SiO_2

We analyze Ge-related defects in Ge-doped SiO_2 using first-principles density functional techniques. Ge is incorporated at the level of ~ 1 mol % and above. The growth conditions of Ge:SiO_2 naturally set up oxygen deficiency, with vacancy concentration increasing by a factor 10^5 over undoped SiO_2, and O vacancies binding strongly to Ge impurities. All the centers considered exhibit potentially EPR-active states, candidates for the identification of the Ge(n) centers. Substitutional Ge produces an apparent gap shrinking via its extrinsic levels.Comment: RevTeX 4 pages, 2 ps figure

Stability of Ge-related point defects and complexes in Ge-doped SiO_2

We analyze Ge-related defects in Ge-doped SiO_2 using first-principles density functional techniques. Ge is incorporated at the level of ~ 1 mol % and above. The growth conditions of Ge:SiO_2 naturally set up oxygen deficiency, with vacancy concentration increasing by a factor 10^5 over undoped SiO_2, and O vacancies binding strongly to Ge impurities. All the centers considered exhibit potentially EPR-active states, candidates for the identification of the Ge(n) centers. Substitutional Ge produces an apparent gap shrinking via its extrinsic levels.Comment: RevTeX 4 pages, 2 ps figure

The impacts of surface conditions on the vapor-liquid-solid growth of germanium nanowires on Si (100) substrate

The impacts of surface conditions on the growth of Ge nanowires on a Si (100) substrate are discussed in detail. On SiO2-terminated Si substrates, high-density Ge nanowires can be easily grown. However, on H-terminated Si substrates, growing Ge nanowires is more complex. The silicon migration and the formation of a native SiO2 overlayer on a catalyst surface retard the growth of Ge nanowires. After removing this overlayer in the HF solution, high-density and well-ordered Ge nanowires are grown. Ge nanowires cross vertically and form two sets of parallel nanowires. It is found that nanowires grew along ?110? direction