Absence of ferromagnetism in Mn- and Co-doped ZnO

Following the theoretical predictions of ferromagnetism in Mn- and Co-doped ZnO, several workers reported ferromagnetism in thin films as well as in bulk samples of these materials. While some observe room-temperature ferromagnetism, others find magnetization at low temperatures. Some of the reports, however, cast considerable doubt on the magnetism of Mn- and Co-doped ZnO. In order to conclusively establish the properties of Mn- and Co-doped ZnO, samples with 6 percent and 2 percent dopant concentrations, have been prepared by the low-temperature decomposition of acetate solid solutions. The samples have been characterized by x-ray diffraction, EDAX and spectroscopic methods to ensure that the dopants are substitutional. All the Mn- and Co-doped ZnO samples (prepared at 400 deg C and 500 deg C) fail to show ferromagnetism. Instead, their magnetic properties are best described by a Curie-Weiss type behavior. It appears unlikely that these materials would be useful for spintronics, unless additional carriers are introduced by some means.Comment: 23 pages, 9 figures. submitted to J. Mater. Chem 200

Energy Efficiency Analysis of the Discharge Circuit of Caltech Spheromak Experiment

The Caltech spheromak experiment uses a size A ignitron in switching a 59-μF capacitor bank (charged up to 8 kV) across an inductive plasma load. Typical power levels in the discharge circuit are ~200 MW for a duration of ~10 μs. This paper describes the setup of the circuit and the measurements of various impedances in the circuit. The combined impedance of the size A ignitron and the cables was found to be significantly larger than the plasma impedance. This causes the circuit to behave like a current source with low energy transfer efficiency. This behavior is expected to be common with other pulsed plasma experiments of similar size that employ an ignitron switch

Boxicity and Cubicity of Product Graphs

The 'boxicity' ('cubicity') of a graph G is the minimum natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes (axis-parallel unit cubes) in $R^k$. In this article, we give estimates on the boxicity and the cubicity of Cartesian, strong and direct products of graphs in terms of invariants of the component graphs. In particular, we study the growth, as a function of $d$, of the boxicity and the cubicity of the $d$-th power of a graph with respect to the three products. Among others, we show a surprising result that the boxicity and the cubicity of the $d$-th Cartesian power of any given finite graph is in $O(\log d / \log\log d)$ and $\theta(d / \log d)$, respectively. On the other hand, we show that there cannot exist any sublinear bound on the growth of the boxicity of powers of a general graph with respect to strong and direct products.Comment: 14 page

Approximation bounds on maximum edge 2-coloring of dense graphs

For a graph $G$ and integer $q\geq 2$, an edge $q$-coloring of $G$ is an assignment of colors to edges of $G$, such that edges incident on a vertex span at most $q$ distinct colors. The maximum edge $q$-coloring problem seeks to maximize the number of colors in an edge $q$-coloring of a graph $G$. The problem has been studied in combinatorics in the context of {\em anti-Ramsey} numbers. Algorithmically, the problem is NP-Hard for $q\geq 2$ and assuming the unique games conjecture, it cannot be approximated in polynomial time to a factor less than $1+1/q$. The case $q=2$, is particularly relevant in practice, and has been well studied from the view point of approximation algorithms. A $2$-factor algorithm is known for general graphs, and recently a $5/3$-factor approximation bound was shown for graphs with perfect matching. The algorithm (which we refer to as the matching based algorithm) is as follows: "Find a maximum matching $M$ of $G$. Give distinct colors to the edges of $M$. Let $C_1,C_2,\ldots, C_t$ be the connected components that results when M is removed from G. To all edges of $C_i$ give the $(|M|+i)$th color." In this paper, we first show that the approximation guarantee of the matching based algorithm is $(1 + \frac {2} {\delta})$ for graphs with perfect matching and minimum degree $\delta$. For $\delta \ge 4$, this is better than the $\frac {5} {3}$ approximation guarantee proved in {AAAP}. For triangle free graphs with perfect matching, we prove that the approximation factor is $(1 + \frac {1}{\delta - 1})$, which is better than $5/3$ for $\delta \ge 3$.Comment: 11pages, 3 figure

Rainbow Connection Number and Radius

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) <= r(r + 2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (Star graph for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) <= rk. It is known that computing rc(G) is NP-Hard [Chakraborty et al., 2009]. Here, we present a (r+3)-factor approximation algorithm which runs in O(nm) time and a (d+3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.Comment: Revised preprint with an extra section on an approximation algorithm. arXiv admin note: text overlap with arXiv:1101.574