Edge Roman domination on graphs

An edge Roman dominating function of a graph $G$ is a function $f\colon E(G) \rightarrow \{0,1,2\}$ satisfying the condition that every edge $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e')=2$. The edge Roman domination number of $G$, denoted by $\gamma'_R(G)$, is the minimum weight $w(f) = \sum_{e\in E(G)} f(e)$ of an edge Roman dominating function $f$ of $G$. This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad stating that if $G$ is a graph of maximum degree $\Delta$ on $n$ vertices, then $\gamma_R'(G) \le \lceil \frac{\Delta}{\Delta+1} n \rceil$. While the counterexamples having the edge Roman domination numbers $\frac{2\Delta-2}{2\Delta-1} n$, we prove that $\frac{2\Delta-2}{2\Delta-1} n + \frac{2}{2\Delta-1}$ is an upper bound for connected graphs. Furthermore, we provide an upper bound for the edge Roman domination number of $k$-degenerate graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic graphs. In addition, we prove that the edge Roman domination numbers of planar graphs on $n$ vertices is at most $\frac{6}{7}n$, which confirms a conjecture of Akbari and Qajar. We also show an upper bound for graphs of girth at least five that is 2-cell embeddable in surfaces of small genus. Finally, we prove an upper bound for graphs that do not contain $K_{2,3}$ as a subdivision, which generalizes a result of Akbari and Qajar on outerplanar graphs

Multi-electrolyte-step anodic aluminum oxide method for the fabrication of self-organized nanochannel arrays

Nanochannel arrays were fabricated by the self-organized multi-electrolyte-step anodic aluminum oxide [AAO] method in this study. The anodization conditions used in the multi-electrolyte-step AAO method included a phosphoric acid solution as the electrolyte and an applied high voltage. There was a change in the phosphoric acid by the oxalic acid solution as the electrolyte and the applied low voltage. This method was used to produce self-organized nanochannel arrays with good regularity and circularity, meaning less power loss and processing time than with the multi-step AAO method

The contributions of qqqqqˉqqqq\bar{q} components to the axial charges of proton and its resonances

We calculate the axial charges of the proton and its resonances in the framework of the constituent quark model, which is extended to include the $qqqq\bar{q}$ components. If 20% admixtures of the $qqqq\bar{q}$ components in the proton are assumed, the theoretical value for the axial charge in our model is in good agreement with the empirical value, which can not be well reproduced in the traditional constituent quark model even though the $SU(6) \bigotimes O(3)$ symmetry breaking or relativistic effect is taken into account. We also predict an unity axial charge for $N^{*}(1440)$ with 30% $qqqq\bar{q}$ components constrained by the strong and electromagnetic decays.Comment: 4 pages, 4 table

Selection of thermodynamic models for combinatorial control of multiple transcription factors in early differentiation of embryonic stem cells

<p>Abstract</p> <p>Background</p> <p>Transcription factors (TFs) have multiple combinatorial forms to regulate the transcription of a target gene. For example, one TF can help another TF to stabilize onto regulatory DNA sequence and the other TF may attract RNA polymerase (RNAP) to start transcription; alternatively, two TFs may both interact with both the DNA sequence and the RNAP. The different forms of TF-TF interaction have different effects on the probability of RNAP's binding onto the promoter sequence and therefore confer different transcriptional efficiencies.</p> <p>Results</p> <p>We have developed an analytical method to identify the thermodynamic model that best describes the form of TF-TF interaction among a set of TF interactions for every target gene. In this method, time-course microarray data are used to estimate the steady state concentration of the transcript of a target gene, as well as the relative changes of the active concentration for each TF. These estimated concentrations and changes of concentrations are fed into an inference scheme to identify the most compatible thermodynamic model. Such a model represents a particular way of combinatorial control by multiple TFs on a target gene.</p> <p>Conclusions</p> <p>Applying this approach to a time-course microarray dataset of embryonic stem cells, we have inferred five interaction patterns among three regulators, Oct4, Sox2 and Nanog, on ten target genes.</p