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In this paper, we study the cycle distribution of random low-density parity-check (LDPC) codes, randomly constructed protograph-based LDPC codes, and random quasi-cyclic (QC) LDPC codes. We prove that for a random bipartite graph, with a given (irregular) degree distribution, the distributions of cycles of different length tend to independent Poisson distributions, as the size of the graph tends to infinity. We derive asymptotic upper and lower bounds on the expected values of the Poisson distributions that are independent of the size of the graph, and only depend on the degree distribution and the cycle length. For a random lift of a bi-regular protograph, we prove that the asymptotic cycle distributions are essentially the same as those of random bipartite graphs as long as the degree distributions are identical. For random QC-LDPC codes, however, we show that the cycle distribution can be quite different from the other two categories. In particular, depending on the protograph and the value of c, the expected number of cycles of length c, in this case, can be either Θ(N ) or Θ(1), where N is the lifting degree (code length). We also provide numerical results that match our theoretical derivations.Our results provide a theoretical foundation for emperical results that were reported in the literature but were not well-justified. They can also be used for the analysis and design of LDPC codes and associated algorithms that are based on cycles.

A proper labeling of a graph is an assignment of integers to some elements of a graph, which may be the vertices, the edges, or both of them, such that we obtain a proper vertex coloring via the labeling subject to some conditions. The problem of proper labeling offers many variants and received a great interest during recent years. We consider the algorithmic complexity of some variants of the proper labeling problems, we present some polynomial time algorithms and $ \mathbf{NP} $-completeness results for them

Graph orientation is a well-studied area of graph theory. A proper orientation of a graph G = (V, E) is an orientation D of E(G) such that for every two adjacent vertices v and u, d −where Γ is the set of proper orientations of G. We have χ(G) − 1 ≤ − → χ (G) ≤ ∆(G). We show that, it is NP-complete to decide whether − → χ (G) = 2, for a given planar graph G. Also, we prove that there is a polynomial time algorithm for determining the proper orientation number of 3-regular graphs. In sharp contrast, we will prove that this problem is NP-hard for 4-regular graphs.

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A new-designed surfactant was formulated to tolerate the harsh conditions of oil reservoirs including high salinity of the formation brine and temperature. The specific emulsion and interfacial tension (IFT) behavior of this new surface active agent were investigated by performing emulsion stability tests, emulsion size analysis, and IFT behavior in the presence of four different types of alkalis. Image processing was utilized to analyze the droplet size distribution using microscopic images of the samples. The results show that depending on the composition of the mixtures, the optimum phase region and interfacial tension behavior change considerably.Solutions containing higher percentage of the surfactant (around 1wt%) show good emulsification capability at different salinities; however, adding any selected alkali to these mixtures reduces the optimum range of salinity tolerance. Mixtures of the surfactant and Triethanolamine exhibit optimum three-phase region at higher salinity conditions compared to other alkaline/surfactant solutions. Increasing the solution salinity reduces the IFT for surfactant solutions however by adding any alkalis the trend was reversed. Considering high values for solubilization-ratio, feasible size of the emulsion droplets, and low IFT values result in promising condition for more oil recovery using a chemical enhanced oil recovery process.

ObjectiveAfghan refugee women in Iran confront many problems in dealing with COVID-19 due to their fragile conditions. Therefore, the aim of this study was to explore the challenges of Afghan refugee women in the face of COVID-19 in Iran with a qualitative approach.MethodsThe present study was conducted with a qualitative approach among Afghan refugee women in Iran. Data were collected through semi-structured face-to-face and telephone interviews and were saturated with 30 women. Both targeted and snowball sampling were used. Data were analyzed using conventional qualitative content analysis and Graneheim and Lundman method. Guba and Lincoln criteria were observed to evaluate the quality of research results.Results143 primary codes, 12 subcategories and five main categories were obtained from data analysis. The main categories include little knowledge and information (limited access to information resources, incomplete knowledge about COVID-19), family challenges (intensified experience of violence and conflict in the family, problems related to childbirth and pregnancy), socio-economic challenges (exacerbation of economic problems, high-risk living conditions, social isolation, limited support of social and health organizations), health issues (problems related to treatment, injustice in providing services and facilities) and problems after the death of a COVID-19 patient (burial challenges for immigrants; lack of funeral rites).ConclusionAfghan refugee women in Iran are very vulnerable facing COVID-19 due to their fragile conditions. Social and health institutions and organizations need to provide more support to these women so that they can protect their health and that of their families against COVID-19 and the damage caused by it.

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