A graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense related. The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line). A directed graph is a graph in which edges have orientation. A simple graph is a graph that does not have more than one edge between any two vertices and no edge starts and ends at the same vertex.  For a simple undirected graph G with order n, and let  denotes its complement. Let δ(G), ∆(G) denotes the minimum degree and maximum degree of G respectively. The complement degree polynomial of G is the polynomial CD[G,x]= , where C

Sickle cell disease (SCD) is a hereditary ailment that can cause severe pain and suffering to people who are affected. However, with continued investment in research and treatment options, we can make progress towards improving the lives of those with SCD. Over 40% of patients experience painful vaso-occlusive crises (VOCs), so we must work towards finding solutions and providing support for those living with this condition, These episodes, a hallmark of SCD, significantly contribute to morbidity, mortality, and a diminished quality of life, while also incurring substantial healthcare costs. Chronic pain particularly affects older adolescents and adults with SCD, with over half reporting daily discomfort. Opioid-based analgesics, though sti

With simple and undirected connected graph Φ, the Schultz and modified Schultz polynomials are defined as  and , respectively, where the summation is taken over all unordered pairs of distinct vertices in V(Φ), where V(Φ) is the vertex set of Φ, degu  is the degree of vertex u and d(v,u) is the ordinary distance between v and u, u≠v. In this study, the Shultz distance, modified Schultz distance, the polynomial, index, and average for both have been generalized, and this generalization has been applied  to some special graphs.

The ring modulator described in part I of this paper is designed here for two operating wavelengths 1550nm and 1310nm. For each wavelength, three structures are designed corresponding to three values of polymer slot widths (40, 50 and 60nm). The performance of these modulators are simulated using COMSOL software (version 4.3b) and the results are discussed and compared with theoretical predictions. The performance of intensity modulation/direct detection short range and long rang optical communication systems incorporating the designed modulators is simulated for 40 and 100Gb/s data rates using Optisystem software (version 12). The results reveal that an average energy per bit as low as 0.05fJ can be obtained when the 1550nm modulator is d

This paper presents comprehensive analysis and investigation for 1550nm and 1310nm ring optical modulators employing an electro-optic polymer infiltrated silicon-plasmonic hybrid phase shifter. The paper falls into two parts which introduce a theoretical modeling framework and performance assessment of these advanced modulators, respectively. In this part, analytical expressions are derived to characterize the coupling effect in the hybrid phase shifter, transmission function of the modulator, and modulator performance parameters. The results can be used as a guideline to design compact and wideband optical modulators using plasmonic technology

F index is a connected graph, sum of the cubes of the vertex degrees. The forgotten topological index has been designed to be employed in the examination of drug molecular structures, which is extremely useful for pharmaceutical and medical experts in understanding the biological activities. Among all the topological indices, the forgotten index is based on degree connectivity on bonds. This paper characterized the forgotten index of union of graphs, join graphs, limits on trees and its complements, and accuracy is measured. Co-index values are analyzed for the various molecular structure of chemical compounds

Background: The ultimate purpose of this prospective study is to estimate and measure swelling associated with surgical extrac¬tion of impacted mandibular third molars in different four post-operative times and to identify the risk factors associated with determination of their risk degree. Material and Methods: In this prospective cohort study 159 consecutive cases in which removal of impacted lower third molars in 107outpatients were evaluated. Five groups of variables have been studied which are regarded as a potential factor for swelling after mandibular third removal which will enable the surgeon to predict and counsel high risk patients in order to offer a preventive strategy. Results: Facial measurements were carried out on 1st, 2

In this paper we generalize Jacobsons results by proving that any integer  in   is a square-free integer), belong to . All units of  are generated by the fundamental unit  having the forms

our generalization build on using the conditions

This leads us to classify the real quadratic fields  into the sets  Jacobsons results shows that  and Sliwa confirm that  and  are the only real quadratic fields in .

An extensive program of laboratory testing was conducted on ring footing rested on gypseous soil brought from the north of Iraq (Salah El-Deen governorate) with a gypsum content of 59%. There are limited researches available, and even fewer have been done experimentally to understand how to ring footings behave; almost all the previous works only concern the behavior of ring footing under vertical loads, Moreover, relatively few studies have examined the impact of eccentric load and inclined load on such footing. In this study, a series of tests, including dry and wet tests, were carried out using a steel container (600×600×600) mm, metal ring footing (100 mm outer diameter and 40 mm inner diameter) was placed in the m

     Graceful labeling of a graph  with q edges is assigned the labels for its vertices by some integers from the set such that no two vertices received the same label, where each edge is assigned the absolute value of the difference between the labels of its end vertices and the resulting edge labeling running from 1 to  inclusive. An edge labeling of a graph G is called vertex anntimagic, if all vertex weights are pairwise distinct, where the vertex weight of a vertex under an edge labeling is the sum of the label of all edges incident with that vertex.  In this paper, we address the problem of finding graceful antimagic labelin for split of the star graph ,  graph,  graph, jellyfish graph , Dragon graph , ki