Independent sets of some graphs associated to commutative rings

Let $G=(V,E)$ be a simple graph. A set $S\subseteq V$ is independent set of $G$, if no two vertices of $S$ are adjacent. The independence number $\alpha(G)$ is the size of a maximum independent set in the graph. %An independent set with cardinality Let $R$ be a commutative ring with nonzero identity and $I$ an ideal of $R$. The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is an undirected graph whose vertices are the nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy = 0$. Also the ideal-based zero-divisor graph of $R$, denoted by $\Gamma_I(R)$, is the graph which vertices are the set {x\in R\backslash I | xy\in I \quad for some \quad y\in R\backslash I\} and two distinct vertices $x$ and $y$ are adjacent if and only if $xy \in I$. In this paper we study the independent sets and the independence number of $\Gamma(R)$ and $\Gamma_I(R)$.Comment: 27 pages. 22 figure

Review of zincblende ZnO: Stability of metastable ZnO phases

Common II-VI compound semiconducting materials are stable thermodynamically with zincblende phase, while the II-O materials such as zinc oxide (ZnO) and beryllium oxide (BeO) are stable with wurtzite phase, and cadmium oxide (CdO) and magnesium oxide (MgO) are stable in rocksalt phase. This phase disharmony in the same material family laid a challenge for the basic physics and in practical applications in optoelectronic devices, where ternary and quaternary compounds are employed. Thermodynamically the zincblende ZnO is a metastable phase which is free from the giant internal electric fields in the [001] directions and has an easy cleavage facet in the ⟨110⟩ directions for laser cavity fabrication that combined with evidence for the higher optical gain. The zincblende materials also have lower ionicity that leads to the lower carrier scattering and higher doping efficiencies. Even with these outstanding features in the zincblende materials, the growth of zincblende ZnO and its fundamental properties are still limited. In this paper, recent progress in growth and fundamental properties of zincblende ZnOmaterial has been reviewed.This research is supported in part by the Australian Research Council, Australia, Institute of Physical and Chemical Research RIKEN, and the Ministry of Education, Science, Sports and Culture, Japan

MEASURING THE PERFORMANCE OF TWO-STAGE PRODUCTION SYSTEMS WITH SHARED INPUTS BY DATA ENVELOPMENT ANALYSIS

As a non-parametric technique in Operations Research and Economics, Data Envelopment Analysis (DEA) evaluates the relative efficiency of peer production systems or decision making units (DMUs) that have multiple inputs and outputs. In recent years, a great number of DEA studies have focused on two-stage production systems in series, where all outputs from the first stage are intermediate products that make up the inputs to the second stage. There are, of course, other types of two-stage processes that the inputs of the system can be freely allocated among two stages. For this type of two-stage production system, the conventional two-stage DEA models have some limitations e.g. efficiency formulation and linearizing transformation. In this paper, we introduce a relational DEA model, considering series relationship among two stages, to measure the overall efficiency of two-stage production systems with shared inputs. The linearity of DEA models is preserved in our model. The proposed DEA model not only evaluates the efficiency of the whole process, but also it provides the efficiency for each of the two sub-processes. A numerical example of US commercial banks from literature is used to clarify the model.Data envelopment analysis, Decision making unit, Two-stage, Shared input, Efficiency

The Wiener, Eccentric Connectivity and Zagreb Indices of the Hierarchical Product of Graphs

Let G1 = (V1, E1) and G2 = (V2, E2) be two graphs having a distinguished or root vertex, labeled 0. The hierarchical product G2 ⊓ G1 of G2 and G1 is a graph with vertex set V2 × V1. Two vertices y2y1 and x2x1 are adjacent if and only if y1x1 ∈ E1 and y2 = x2; or y2x2 ∈ E2 and y1 = x1 = 0. In this paper, the Wiener, eccentric connectivity and Zagreb indices of this new operation of graphs are computed. As an application, these topological indices for a class of alkanes are computed. ACM Computing Classification System (1998): G.2.2, G.2.3.The research of this paper is partially supported by the University of Kashan under grant no 159020/12

Strongyloides stercoralis: The Most Prevalent Parasitic Cause of Eosinophilia in Gilan Province, Northern Iran

Background: Eosinophilia occurs in a wide variety of situations such as parasitic infections, aller­gic disorders, and malignancies. Most cases of eosinophilia of parasitic origin, especially those with a tissue migration life cycles consists of human infections by helminth parasites. The aim of present study was to determine the parasitic causes of eosinophilia in patients in a major endemic area of human fascioliasis in Gilan Province, northern part of Iran.Methods: One hundred and fifty patients presenting with an elevated eosinophilia attending infec­tious disease clinics with or without clinical symptoms, were examined. After clinical his­tory evaluation and physical examination, coprological examinations were performed using the formalin-ether and the Kato-Katz techniques for detection of Fasciola sp. and intestinal parasites.Results: Forty two percent of patients were infected with S. stercoralis, nine (6%) were found to be infected with Fasciola sp. while only a single patient (0.7%) were infected by Ttrichostrongy­lus sp.Conclusion: Local clinicians in Gilan may consider eosinophilia as a suggestive indication for diagnosis of human fascioliasis, especially when microscopic stool and/or serological tests are negative. Based on the results, local physicians should consider S. stercoralis as the potential causes of eosinophilia in patients with elevated eosinophilia

The Eccentric Connectivity Polynomial of some Graph Operations

The eccentric connectivity index of a graph G, ξ^C, was proposed by Sharma, Goswami and Madan. It is defined as ξ^C(G) = ∑ u ∈ V(G) degG(u)εG(u), where degG(u) denotes the degree of the vertex x in G and εG(u) = Max{d(u, x) | x ∈ V (G)}. The eccentric connectivity polynomial is a polynomial version of this topological index. In this paper, exact formulas for the eccentric connectivity polynomial of Cartesian product, symmetric difference, disjunction and join of graphs are presented.* The work of this author was supported in part by a grant from IPM (No. 89050111)