Pathos degree prime graph of a tree

Let T be a tree of order n (n ≥ 2). A pathos degree prime graph of T, written PDP(T), is a graph whose vertices are the vertices and paths of a pathos of T, with two vertices of PDP(T) adjacent whenever the degree of the corresponding vertices of T are unequal and relatively prime; or the corresponding paths Pʹᵢ and Pʹᵢ (i ≠ j) of a pathos of T have a vertex in common; or one corresponds to the path Pʹ and the other to a vertex v and Pʹ begins (or ends) at v such that v is a pendant vertex in T. We look at some properties of this graph operator. For this class of graphs we discuss the planarity; outerplanarity; maximal outerplanarity; minimally nonouterplanarity; Eulerian; and Hamiltonian properties these graphs.Publisher's Versio

Totally magic d-lucky number of graphs

In this paper we introduce a new labeling named as, totally magic d-lucky labeling, find the totally magic d-lucky number of some standard graphs like wheel, cycle, bigraph etc. and find the totally magic d-lucky number of some zero divisor graphs. A totally magic d-lucky labeling  of a graph G = (V, E) is a labeling of vertices and label the graph's edges using the total label of its incident vertices in such a way that for any two different incident vertices u and v, their colors ,  are distinct and for any different edges in a graph, their weights    are same Where  represents the degree of u in a graph and  represents the open neighbourhood of u in a graph

Some Topics of Special Interest in Graph Theory

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Discussion On Some Important Topics in Graph Theory

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Some Investigations in the Theory of Graphs

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Persistence Motivation of Foreign-born Doctoral Recipients in the Field of Education

Although there has been growing enrollment and doctoral degree production of foreign-born doctoral students in U.S. higher education, persistence/degree completion and time-to-degree remain a continuing problem in doctoral education in general. Despite the substantial number of studies conducted on various aspects of doctoral education, there is still a scarcity of research on exploring the doctoral process of foreign-born students. When foreign-born students are included in the samples, researchers use a theoretical framework that does not give a comprehensive understanding of doctoral experiences of foreign-born students thereby ignoring the salient differences between them and their native-born counterparts, which makes it difficult for U.S. graduate schools to respond to and identify the distinctive needs of this growing group of doctoral students. Also, the field of education has continued to experience the longest time-to-degree in American higher education, with the median duration between starting and completing graduate school from 10.7 to 12.7 years compared to 7.7 to 7.9 years in all fields including education. This study explored the factors that motivate foreign-born doctoral recipients to pursue and persist toward the completion of their doctorate in the field of education. Using expectancy-value theory and socialization theory as theoretical perspectives, particular attention was paid to how expectancies and values placed on earning a doctorate motivated foreign-born doctoral recipients to pursue their doctoral degree and the strategies they used to mitigate the costs they experienced while in the program, as well as how socialization elements may have contributed to participants’ persistence toward degree completion

Linking Metabolic Capacity and Molecular Biology of Methylocystis sp. Strain SC2 by a Newly Developed Proteomics Workflow

Microbial methane oxidation is one of the fundamental processes in global methane cycle. Methane-oxidizing bacteria, or methanotrophs, are the major biological sink for the methane produced from anthropogenic and natural sources. Our model organism, Methylocystis sp. strain SC2, is one of the best-studied representatives of alphaproteobacterial (type IIa) methanotrophs. Proteobacterial methanotrophs possess a unique cell architecture characterized by intracytoplasmic membranes (ICMs). The cellular amount of the ICMs is increasing with methanotrophic activity. The presence of ICMs makes molecular biology approaches, but in particular global proteomics, highly challenging. In this study, we therefore aimed to develop an efficient proteomics workflow for strain SC2 and to apply this state-of-the-art tool for investigation of the strain SC2 response to environmental factors. To successfully develop the proteomics workflow, we particularly focused on an efficient solubilization and digestion of the integral membrane proteins of strain SC2 for further downstream analysis. We introduced the so-called crude-MS proteomics workflow, upon assessing and optimizing all the major steps in the proteomics workflow, including cell lysis, protein solubilization, and protein digestion. Our new SC2 proteomics workflow greatly increased not only the protein quantification accuracy (mean coefficient of variation 3.2 %) but also the proteome coverage to 62%, with up to 10-fold increase in the detection intensity of membrane-associated proteins. Previous studies have shown that the LysC/trypsin tandem digestion resulted in higher coverage of fully cleaved tryptic peptides than a trypsin-only digestion. Therefore, the development of our optimized proteomics workflow involved the application of the LysC/trypsin tandem digestion in detergent environment to increase the SC2 proteome coverage. Prior to publication of our crude-MS approach, all systematic assessments of LysC/trypsin proteolysis were conducted in chaotropic environments, like urea. As a spin-off, we therefore initiated a follow-up study to compare the efficiency of the LysC/trypsin tandem digestion in detergent environments (e.g., SDC, SLS) relative to chaotropic environments. The study revealed that the LysC/trypsin tandem digestion could be efficiently carried out not only in chaotropic environments but also in MS-compatible detergent environments. In fact, the LysC/trypsin tandem digestion in both environments resulted in a higher coverage of fully cleaved peptides than the trypsin-only digestion. After successful development of the crude-MS proteomics workflow, we used this high-throughput method to assess the molecular response of strain SC2 to the availability of hydrogen as a potentially alternative energy source. Starting point of this research was the knowledge that strain SC2 and other Methylocystis spp. possess the genetic potential to produce various hydrogenases. In fact, the addition of 2% hydrogen to the headspace atmosphere led, under limiting concentrations of methane and oxygen, to the complete hydrogen consumption by strain SC2. Concurrently, the SC2 biomass yield was significantly increased, while the methane consumption rate was significantly decreased. Global proteome analyses revealed that the addition of hydrogen induced an increase in the production of Group 1d and Group 2b [NiFe]-hydrogenases, and hydrogenase accessory proteins. Notably, the upregulation of the Group 1d, 2b [NiFe]-hydrogenases was concomitantly linked to a reconstruction of the energy metabolism in strain SC2. In another project, genome-scale metabolic modeling and growth experiments were applied to show that strain SC2 has the capacity to utilize acetate through the glyoxylate assimilation pathway. In addition, the study revealed that in type II methanotrophs, energy demand for methane oxidation is covered by complex I of the electron transport chain. In summary, our research demonstrates how to experimentally link the metabolic potential of Methylocystis sp. strain SC2 with the underlying proteome complexity. Thus, the newly developed highly reproducible SC2 proteomics workflow represents a high-throughput method that makes it possible to achieve in future research an understanding of the molecular adaptation mechanisms of strain SC2 to environmental change

SIMULATING SEISMIC WAVE PROPAGATION IN TWO-DIMENSIONAL MEDIA USING DISCONTINUOUS SPECTRAL ELEMENT METHODS

We introduce a discontinuous spectral element method for simulating seismic wave in 2- dimensional elastic media. The methods combine the flexibility of a discontinuous finite element method with the accuracy of a spectral method. The elastodynamic equations are discretized using high-degree of Lagrange interpolants and integration over an element is accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix and the use of discontinuous finite element method makes the calculation can be done locally in each element. Thus, the algorithm is simplified drastically. We validated the results of one-dimensional problem by comparing them with finite-difference time-domain method and exact solution. The comparisons show excellent agreement