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Abstract Sierpiński graphs are frequently related to fractals, and fractals apply in several fields of science, i.e., in chemical graph theory, computer networking, biology, and physical sciences. Functions and polynomials are powerful tools in computer mathematics for predicting the features of networks. Topological descriptors, frequently graph constraints, are absolute values that characterize the topology of a computer network. In this essay, Firstly, we compute the M-polynomials for Sierpiński-type fractals. We derive some degree-dependent topological invariants after applying algebraic operations on these M-polynomials.

The cactus graph has many practical applications, particularly in radio communication systems. Let $ G = (V, E) $ be a finite, undirected, and simple connected graph, then the edge metric dimension of $ G $ is the minimum cardinality of the edge metric generator for $ G $ (an ordered set of vertices that uniquely determines each pair of distinct edges in terms of distance vectors). Given an ordered set of vertices $ \mathcal{G}_e = \{g_1, g_2, ..., g_k \} $ of a connected graph $ G $, for any edge $ e\in E $, we referred to the $ k $-vector (ordered $ k $-tuple), $ r(e|\mathcal{G}_e) = (d(e, g_1), d(e, g_2), ..., d(e, g_k)) $ as the edge metric representation of $ e $ with respect to $ G_e $. In this regard, $ \mathcal{G}_e $ is an edge metric generator for $ G $ if, and only if, for every pair of distinct edges $ e_1, e_2 \in E $ implies $ r (e_1 |\mathcal{G}_e) \neq r (e_2 |\mathcal{G}_e) $. In this paper, we investigated another class of cacti different from the cacti studied in previous literature. We determined the edge metric dimension of the following cacti: $ \mathfrak{C}(n, c, r) $ and $ \mathfrak{C}(n, m, c, r) $ in terms of the number of cycles $ (c) $ and the number of paths $ (r) $.

The molecular energy, which is the sum of all eigenvalues, is crucial in determining the total π-electron energy of conjugated hydrocarbon molecules. We used machine learning techniques to calculate the energy, inertia, nullity, signature, and Estrada index of molecular graphs for bismuth tri-iodide and benzene rings embedded in P-type surfaces within 2D networks. We applied MATLAB to extract the actual eigenvalues from the data and developed general equations for these molecular properties. We then used these equations to estimate the values and compared them to the actual values through graphical analysis. Our results demonstrate the potential of data-driven techniques in predicting molecular properties and enhancing our understanding of spectral theory.

Abstract Metal–organic frameworks (MOFs) are vital in modern material science, offering unique properties for gas storage, catalysis, and drug delivery due to their highly porous and customizable structures. Chemical graph theory emerges as a critical tool, providing a mathematical model to represent the molecular structure of these frameworks. Topological indices/molecular descriptors are mathematical formulations applied to molecular models, enabling the analysis of physicochemical properties and circumventing costly lab experiments. These descriptors are crucial for quantitative structure-property and structure-activity relationship studies in mathematical chemistry. In this paper, we study the different molecular descriptors of tetracyanobenzene metal–organic framework. We also give numerical comparison of computed molecular descriptors.

Distributed Generation (DG) studies are generally conducted considering a single type of DG units integrated with the Distribution System (DS). However, these studies may not evaluate optimum benefits offered by the DG integration. This paper presents a novel framework for individual and simultaneous allocation of different types of DG units in DS while considering varying load demand and probabilistic generation of DG units. A Salp Swarm Algorithm (SSA) and Particle Swarm Optimization (PSO) based new hybrid method that utilizes a single controlling parameter is proposed for simultaneous allocation of DG units, aiming to minimize a multi-objective index. The strength of the proposed SSA-PSO hybrid approach is validated on seventeen benchmark functions and the performance of the novel framework for optimal allocation of DG units is validated through implementation on the 69-bus system. The results demonstrate that the proposed hybrid approach performs better as compared to other meta-heuristics techniques. Moreover, simultaneous allocation significantly reduces the multi-objective index as compared to individual DG units’ integration.

Graph theory is a crucial branch of mathematics in fields like network analysis, molecular chemistry, and computer science, where it models complex relationships and structures. Many indices are used to capture the specific nuances in these structures. In this paper, we propose a new index, the weighted asymmetry index, a graph-theoretic metric quantifying the asymmetry in a network using the distances of the vertices connected by an edge. This index measures how uneven the distances from each vertex to the rest of the graph are when considering the contribution of each edge. We show how the index can capture the intrinsic asymmetries in diverse networks and is an important tool for applications in network analysis, optimization problems, social networks, chemical graph theory, and modeling complex systems. We first identify its extreme values and describe the corresponding extremal trees. We also give explicit formulas for the weighted asymmetry index for path, star, complete bipartite, complete tripartite, generalized star, and wheel graphs. At the end, we propose some open problems.

This work explored Phyla nodiflora (L.) Greene as a potential source of the bioactive medicinal agent. In this aspect, methanol (PN-M) and dichloromethane (PN-D) extracts were prepared from the whole plant and evaluated for phytochemical composition (total bioactive contents, UHPLC-MS analysis, and HPLC-PDA polyphenolic quantification), biological (antioxidant and enzyme inhibition) potential and in-vivo toxicity. The PN-M was found to contain higher phenolic (26.08 mg GAE/g extract) and flavonoid (50.25 mg QE/g extract) contents which might correlate to the higher radical scavenging (DPPH: 52.94 mg TE/g extract; ABTS: 72.11 mg TE/g extract) and reducing power (FRAP: 71.96 mg TE/g extract; CUPRAC: 142.65 mg TE/g extract) antioxidant potential, as well as AChE (4.33 mg GALAE/g extract), tyrosinase (125.36 mg KAE/g extract), and amylase (1.86 mmol ACAE/g extract) inhibition activity of this extract. In contrast, the PN-D extract was found to be most active for phosphomolybdenum (1.30 mg TE/ g extract) and metal chelation (54.84 mg EDTAE/g extract) assays in addition to BChE (4.70 mg GALAE/g extract) and glucosidase (0.62 mmol ACAE/g extract) enzyme inhibition activity. The PN-M extract on UHPLC-MS analysis revealed the tentative identification of 24 different secondary metabolites, most of which belonged to the flavonoid, glycoside, and terpenoid classes of phytochemicals. The polyphenolic composition of the extracts was appraised by HPLC-PDA. Seven phenolic compounds were identified in the extracts. PN-M was found to be rich in catechin (0.25 µg/extract) and 3-OH benzoic acid (0.64 µg/extract), while PN-D contained epicatechin (0.30 µg/extract), 3-OH-4-MeO benzaldehyde (0.21 µg/extract), and 2,3-Di-Meo benzoic acid (0.97 µg/extract) in higher amounts. The methanol extract was found to be non-toxic even at higher doses. Furthermore, the relationship between the phytochemicals and the tested enzymes was highlighted by molecular docking studies. In sum, this research showed that the studied extracts were effective as enzyme inhibitors and antioxidants, suggesting it would be worth investigating in more depth for further advanced studies to explore its pharmacological properties.

Background: Oral anticancer drugs (OACDs) have been used since the 1950s and are expected to be linked with enhanced life quality which helps patients receive treatment at home. Although more convenient in use than intravenous or other infused drugs, the high costs of these OACDs have been proven controversial. Methods: The literature was searched systematically from PubMed, Cochrane, Embase, and Scopus according to the PRISMA guidelines. The inclusion criteria included studies in the English language, evaluating Pharmacoeconomics, and evaluating cost-utility and cost-effectiveness related to OACDs. The information on the included studies was synthesized in the form of summary tables. Results: Thirteen studies were included for quantitative analysis, which evaluated the costs or cost-effectiveness of different OACDs used for various types of cancer. It was found that the average cost for the OACDs was $80979/year in attaining (quality-adjusted life per year) QALY across different countries. The costs of different OACDs were highly varied alone in the US. The included studies' results were highly varied, limiting the findings' interpretations. Conclusion: Various studies about cost-effectiveness persist insufficiently represented in the literature regarding OACDs, suggesting the requirement of more cost-effectiveness analysis shortly. The increased costs of these OACDs require careful evaluation of the cost-effectiveness studies.

For a simple connected graph $ G = (V, E) $, a vertex $ x\in V $ distinguishes two elements (vertices or edges) $ x_1\in V, y_1 \in E $ if $ d(x, x_1)\neq d(x, y_1). $ A subset $ Q_m\subset V $ is a mixed metric generator for $ G, $ if every two distinct elements (vertices or edges) of $ G $ are distinguished by some vertex of $ Q_m. $ The minimum cardinality of a mixed metric generator for $ G $ is called the mixed metric dimension and denoted by $ dim_m(G). $ In this paper, we investigate the mixed metric dimension for different families of ladder networks. Among these families, we consider Möbius ladder, hexagonal Möbius ladder, triangular Möbius ladder network and conclude that all these families have constant-metric, edge metric and mixed metric dimension.

The UAE's environment is characterized by low rainfall, high temperatures, poor soil and lack of natural waterways. The country has been making tremendous efforts to build a sustainable agricultural and animal wealth sector that is better able to contribute to its national agenda of food security. Commercial cattle farming in such challenging conditions would require an effective disease prediction and warning system for optimizing the production dynamics. Therefore, we are proposing the use of smart livestock system comprising of orally ingested bio-capsules for monitoring the livestock health, connected through a customized long-range network (430 MHz) in the United Arab Emirates, which monitors the real time animal body condition data (temperature and body activity) using a mobile or web-based application. The study demonstrated a 100% success rate in estrus alarms in the test animals, 61.29% for calving alerts and an accurate disease prediction warning in all the animals diagnosed unhealthy. the conception rate in these estrus diagnosed animals with artificial insemination was 61.53%, which was significantly higher than the conception rate in the absence of this detection technology at the farm. The capsule was well tolerated in all the test animals and did not show any acute adverse or allergic reactions. This is the first of its kind of study using these advanced bio-capsules and development of the customized long-range network in the United Arab Emirates for smart livestock healthcare management.

Let $ \psi = (V, E) $ be a simple connected graph. The distance between $ \rho_1, \rho_2\in V(\psi) $ is the length of a shortest path between $ \rho_1 $ and $ \rho_2. $ Let $ \Gamma = \{\Gamma_1, \Gamma_2, \dots, \Gamma_j\} $ be an ordered partition of the vertices of $ \psi $. Let $ \rho_1\in V(\psi) $, and $ r(\rho_1|\Gamma) = \{d(\rho_1, \Gamma_1), d(\rho_1, \Gamma_2), \dots, d(\rho_1, \Gamma_j)\} $ be a $ j $-tuple. If the representation $ r(\rho_1|\Gamma) $ of every $ \rho_1\in V(\psi) $ w.r.t. $ \Gamma $ is unique then $ \Gamma $ is the resolving partition set of vertices of $ \psi $. The minimum value of $ j $ in the resolving partition set is known as partition dimension and written as $ pd(\psi). $ The problem of computing exact and constant values of partition dimension is hard so one can compute bound for the partition dimension of a general family of graph. In this paper, we studied partition dimension of the some families of convex polytopes with pendant edge such as $ R_n^P $, $ D_n^p $ and $ Q_n^p $ and proved that these graphs have bounded partition dimension.

Abstract Background Coronavirus disease 2019 (COVID-19) caused by the novel coronavirus-infected millions globally. Despite a wide range of advised options for the treatment of COVID-19, a single strategy to tackle this pandemic remains elusive, thus far. That is why we are conducting a clinical trial to find out the efficacy of iodine complex to clear a viral load of severe respiratory syndrome coronavirus-2 (SARS-CoV-2) along with a reduction in time taken to alleviate symptoms. Method The proposed study is a placebo-controlled, add-on, randomized trial using parallel group designs. This is a closed-label and adaptive with sample size reassessment, multi-centered design with a 1:1:1:1 allocation ratio and superiority framework. It will be conducted in Shaikh Zayed Post-Graduate Medical Complex, Ali Clinic, and Doctors Lounge, Lahore, Pakistan. This study will have three arms of mild to moderately symptomatic COVID-19 patients (50 patients in each) which will receive ionic-iodine polymer complex with 200 mg of elemental iodine: interventional arm A will have encapsulated, arm B will receive suspension syrup form, arm C will get throat spray, while arm X will be standard care with placebo. Data will be collected on self-constructed, close-ended questionnaires after obtaining written consent. Data will be analyzed using SAS version 9.4. COVID-19 patients will be monitored by RT-PCR and HRCT (high-resolution computed tomography) chest. In addition to these, the duration of the symptomatic phase and mortality benefits will be analyzed in both groups. Discussion The study is designed to measure the superior efficacy of the iodine complex as an add-on in treating COVID-19-positive patients with mild to moderate symptoms. This combination is hypothesized to improve various parameters like rapid viral load reduction, clinical and radiological improvement, lower mortality, and reduction in hospitalization. The trial will aid in devising a better strategy to cope with COVID-19 in a relatively inexpensive and accessible way. The implications are global, and this could prove itself to be the most manageable intervention against COVID-19 especially for patients from limited-resource countries with deprived socioeconomic status. Trial registration ClinicalTrials.gov NCT04473261 . Registered on July 16, 2020.

Optimal graph embeddings represent graphs in a lower dimensional space in a way that preserves the structure and properties of the original graph. These techniques have wide applications in fields such as machine learning, data mining, and network analysis. Do we have small (if possible minimal) k-connected graphs with the property that for any j vertices there is a longest path avoiding all of them? This question of Zamfirescu (1972) was the first variant of Gallai’s question (1966): Do all longest paths in a connected graph share a common vertex? Several good examples answering Zamfirescu’s question are known. In 2001, he asked to investigate the family of geometrical lattices with respect to this property. In 2017, Chang and Yuan proved the existence of such graphs in Archimedean tiling. Here, we prove that the graphs presented by Chang and Yuan are not optimal by constructing such graphs of sufficiently smaller orders. The problem of finding nonconcurrent longest paths in Archimedean tessellations refers to finding paths in a lattice such that the paths do not overlap or intersect with each other and are as long as possible. The complexity of embedding graph is still unknown. This problem can be challenging because it requires finding paths that are both long and do not intersect, which can be difficult due to the constraints of the lattice structure.

Early detection, diagnosis and treatment of cattle disease is very important for any livestock farming enterprise, to remain viable. Improved cattle health and productivity through advanced and innovative smart farming technologies will pave the way for the United Arab Emirates in achieving its national agenda of food security. Advanced livestock health care bio-capsules were administered through mouth into the rumen of cattle in a private dairy farm to a group of cattle to monitor their health. The effect of using such biosensors on the animal health was not previously described by any other researchers. Hence, in the current study, we tried to understand the effects of the bio-capsules on the cattle health, if any. This was done through collecting and screening blood samples from the test animals for different hemato-biochemical parameters and our results showed that there were no harmful effects of using such bio-capsules for livestock health monitoring.

Deep Neural Networks and Optimization Algorithms have many applications in engineering problems and scientific research [...]

Let G = (V(G), E(G)) be a graph with no loops, numerous edges, and only one component, which is made up of the vertex set V(G) and the edge set E(G). The distance d(u, v) between two vertices u, v that belong to the vertex set of H is the shortest path between them. A k-ordered partition of vertices is defined as β = {β1, β2, …, βk}. If all distances d(v, βk) are finite for all vertices v ∈ V, then the k-tuple (d(v, β1), d(v, β2), …, d(v, βk)) represents vertex v in terms of β, and is represented by r(v|β). If every vertex has a different presentation, the k-partition β is a resolving partition. The partition dimension of G, indicated by pd(G), is the minimal k for which there is a resolving k-partition of V(G). The partition dimension of Toeplitz graphs formed by two and three generators is constant, as shown in the following paper. The resolving set allows obtaining a unique representation for computer structures. In particular, they are used in pharmaceutical research for discovering patterns common to a variety of drugs. The above definitions are based on the hypothesis of chemical graph theory and it is a customary depiction of chemical compounds in form of graph structures, where the node and edge represent the atom and bond types, respectively.

This study was designed to seek the phytochemical analysis, antioxidant, enzyme inhibition, and toxicity potentials of methanol and dichloromethane (DCM) extracts of aerial and root parts of Crotalaria burhia. Total bioactive content, high-performance liquid chromatography-photodiode array detector (HPLC-PDA) polyphenolic quantification, and ultra-high performance liquid chromatography-mass spectrometry (UHPLC-MS) analysis were utilized to evaluate the phytochemical composition. Antioxidant [including 2,2-diphenyl-1-picryl-hydrazyl-hydrate (DPPH)], 2,2′-azino-bis[3-ethylbenzothiazoline-6-sulfonic acid (ABTS), ferric reducing antioxidant power assay (FRAP), cupric reducing antioxidant capacity CUPRAC, phosphomolybdenum, and metal chelation assays] and enzyme inhibition [against acetylcholinesterase (AChE), butyrylcholinesterase (BChE), α-glucosidase, α-amylase, and tyrosinase] assays were carried out for biological evaluation. The cytotoxicity was tested against MCF-7 and MDA-MB-231 breast cell lines. The root-methanol extract contained the highest levels of phenolics (37.69 mg gallic acid equivalent/g extract) and flavonoids (83.0 mg quercetin equivalent/g extract) contents, and was also the most active for DPPH (50.04 mg Trolox equivalent/g extract) and CUPRAC (139.96 mg Trolox equivalent /g extract) antioxidant assays. Likewise, the aerial-methanol extract exhibited maximum activity for ABTS (94.05 mg Trolox equivalent/g extract) and FRAP (64.23 mg Trolox equivalent/g extract) assays. The aerial-DCM extract was noted to be a convincing cholinesterase (AChE; 4.01 and BChE; 4.28 mg galantamine equivalent/g extract), and α-glucosidase inhibitor (1.92 mmol acarbose equivalent/g extract). All of the extracts exhibited weak to modest toxicity against the tested cell lines. A considerable quantities of gallic acid, catechin, 4-OH benzoic acid, syringic acid, vanillic acid, 3-OH-4-MeO benzaldehyde, epicatechin, p-coumaric acid, rutin, naringenin, and carvacrol were quantified via HPLC-PDA analysis. UHPLC-MS analysis of methanolic extracts from roots and aerial parts revealed the tentative identification of important phytoconstituents such as polyphenols, saponins, flavonoids, and glycoside derivatives. To conclude, this plant could be considered a promising source of origin for bioactive compounds with several therapeutic uses.

Sierpiński networks are the most studied networks of fractal nature with applications in various fields of science. A generalized Sierpiński network is obtained by copying the base network, resulting in the self-similar network. The extended Sierpiński networks are obtained by introducing a new vertex in a generalized Sierpiński network and attaching this vertex with the extreme vertices. Certain network invariants are used to find thermodynamic properties, physio-chemical properties, and biological activities of chemical compounds. These network invariants play a dynamic role in QSAR/QSPR study. In this paper, we discussed Zagreb indices and forgotten topological index for extended Sierpiński networks by using any base network H. Moreover, for the studied topological indices, we attained some bounds using different parameters i.e. order, size, maximum and minimum degrees of vertices in network H.

Abstract Electrical vehicle (EV) technology has gained popularity due to its higher efficiency, less maintenance, and lower dependence on fossil fuels. However, a longer charging time is a significant barrier to its complete adaptation. Solid state transformer (SST) based extreme fast charging schemes have emerged as an appealing idea with an ability to provide a refuelling capability analogous to that of gasoline vehicles. Therefore, this paper reviews the EV charger requirements, specifications, and design criteria for high power applications. At first, the key barriers of using a traditional low frequency transformer (LFT) are discussed, and potential solutions are suggested by replacing the conventional LFT with high frequency SST at extreme fast‐charging (XFC) stations. Then, various SST‐based converter topologies and their control for EV fast‐charging stations are described. The reviewed control strategies are compared while considering several factors such as harmonics, voltage drop under varying loading conditions, dc offset load unbalances, overloads, and protection against system disturbances. Furthermore, the realization of SST for EV charging is comprehensively discussed, which facilitates understanding the current challenges, based on which potential solutions are also suggested.

The partition dimension is the most complicated resolving parameter of all the resolvability parameters. In this parameter, the vertex set of a graph is completely subdivided into subsets so that each vertex of a graph has a unique identification in terms of the partitioning of its selected subsets. In this paper, we consider Carbon nanocone structure (CNC) more specifically one pentagonal carbon nanocone CNCζ,5. It is also regarded as a useful structure in terms of various applications. To delve deeper into this structure, we provide a graphical representation of it and identified sharp bounds for the partition dimension. We conclude that the pentagonal carbon nanocone CNCζ,5 has partition dimension ≤4.

The measure or determination of uncertainty of a system is known as entropy. After the rich idea of entropy, it gains attraction and interest in graph theory, due to its applications in applied mathematical chemistry. First time the idea graph entropy was used in 1955, to know the structural information of chemical networks or graphs. In mathematical chemistry, structures are the build-up of atoms/vertices and bonds/edges. Carbon nanotubes are also very famous since they have been used for memory devices and tissue engineering. This productive chemical structure is studied in this paper by inserting a cap on one and both ends. We measure the entropy of these three topologies of carbon nanotube. Some comparative work is also given, along with the analytical study of entropy measures of armchair carbon nanotubes.

Distributed Generation (DG) based on Renewable Energy Sources (RES) are considered as an effective and economical technology for the advancement of an Electric Power System (EPS) to fulfill the load demand. Mostly, studies pertaining to DG planning are performed while considering constant load demand and DG generation. However, these considerations may provide misleading and inconsistent values for loss reduction, voltage profile, power quality, and other operational parameters. Therefore, this paper proposes a novel framework to determine the impact of different Time Varying Voltage Dependent (TVVD) load models on wind DG planning study. Firstly, wind DG optimal allocation is performed using Salp Swarm Algorithm (SSA) for different TVVD load models. Afterwards, impact of different TVVD load models on wind DG planning is investigated. Comparative evaluation of various impact indices, real and reactive power (losses and intakes), penetration level, and apparent power support provided due to integration of wind DG are discussed for various TVVD loads. The analysis of results indicates that TVVD loads have a significant impact on performance of distribution system and DG planning studies.

Silica comes in three different crystalline forms, with quartz being the most common and plentiful in the crust of our planet. Other variations are created when quartz is heated. Each chemical structure may be deduced from graphs in which atoms alternate as vertices and edges as bonds, according to chemical graph theory. The latest advanced topic of resolvability parameters of a graph is, where the complete topology is formed in a particular way that every single atom’s unique location is obtained. The resolving basis, edge resolving basis, and some generalizations were investigated in this paper as resolvability characteristics of Quartz. Understanding and dealing with structures is made easier by transforming the entire topology into a unique form given by resolvability parameters.

The development of highly efficient models of Photovoltaic (PV) cells and modules is essential for optimized performance, evaluation and control of solar PV systems. The accurate estimation of PV cells parameters is a challenging task because of their non-linear characteristics. In this paper, an improved variant of Flower Pollination Algorithm (FPA) is proposed for accurate estimation of PV cells and modules parameters. The proposed algorithm involves double exponential based dynamic switch probability and a dynamic step size function that mitigate the limitations of conventional FPA. The dynamic switch probability improves the overall performance of algorithm by maintaining a balance between local and global search, while dynamic step function controls the search speed which avoids premature convergence and local optima stagnation. Moreover, Newton Raphson Method is utilized for accurate computation of estimated current for optimum set of estimated parameters. The proposed methodology is evaluated using seven benchmark functions and three case studies; 1- RTC France silicon PV cell, 2- Photo-watt PWP-201 PV module and 3- a practical solar PV system (EAGLE PERC 60M 310W monocrystalline PV module) under different environmental conditions by estimating parameters for single and double diode models. The analysis of results indicates that, the proposed approach improves the convergence speed, precision, avoids premature convergence and stagnation in local optima of conventional FPA. Furthermore, comparative analysis of results illustrates that, the proposed approach is more reliable and efficient than many other techniques in literature.

With the tremendous demand for connectivity anywhere and anytime, existing network architectures should be modified. To cope with the challenges that arise due to the increasing flood of devices/users and a diverse range of application requirements, new technologies and concepts must be integrated to enable their benefits. Service providers and business companies are looking for new areas of research to enhance overall system performance. This article gives a detailed survey about the recent 5G technologies, the solutions they provide, and the effect caused by their addition to current cellular networks. It is based on the three most important 5G concepts: Device to Device (D2D), Network Slicing (NS), and Mobile Edge Computing (MEC). This study proposes to design the future 5G networks by the integration of all three technologies. It is believed that spectrum efficiency, energy efficiency, and overall throughput will be greatly improved by using D2D. The system delay and computational load will be reduced as tasks will be handled by edge routers located at the base stations. Thus offloading the core network and the system capital expenses and operational expenses will be reduced significantly by slicing the network.

Complex networks are not easy to decode and understand to work on it, similarly, the Möbius structure is also considered as a complex structure or geometry. But making a graph of every complex and huge structure either chemical or computer-related networks becomes easy. After making easy of its construction, recognition of each vertex (node or atom) is also not an easy task, in this context resolvability parameters plays an important role in controlling or accessing each vertex with respect to some chosen vertices called as resolving set or sometimes dividing entire cluster of vertices into further subparts (subsets) and then accessing each vertex with respect to build in subsets called as resolving partition set. In these parameters, each vertex has its own unique identification and is easy to access despite the small or huge structures. In this article, we provide a resolving partition of hexagonal Möbius ladder graph and discuss bounds of partition dimension of hexagonal Möbius ladder network.

Shannon entropy, also known as information entropy or entropy, measures the uncertainty or randomness of probability distribution. Entropy is measured in bits, quantifying the average amount of information required to identify an event from the distribution. Shannon’s entropy theory initiates graph entropies and develops information-theoretic magnitudes for structural computational evidence of organic graphs and complex networks. Graph entropy measurements are valuable in several scientific fields, such as computing, chemistry, biology, and discrete mathematics. In this study, we investigate the entropy of fractal-type networks by considering cycle, complete, and star networks as base graphs using degree-based topological indices.

Graphs are useful for analysing the structure models in computer science, operations research, and sociology. The word metric dimension is the basis of the distance function, which has a symmetric property. Moreover, finding the resolving set of a graph is NP-complete, and the possibilities of finding the resolving set are reduced due to the symmetric behaviour of the graph. In this paper, we introduce the idea of the edge-multiset dimension of graphs. A representation of an edge is defined as the multiset of distances between it and the vertices of a set, B⊆V(Γ). If the representation of two different edges is unequal, then B is an edge-multiset resolving a set of Γ. The least possible cardinality of the edge-multiset resolving a set is referred to as the edge-multiset dimension of Γ. This article presents preliminary results, special conditions, and bounds on the edge-multiset dimension of certain graphs. This research provides new insights into structure models in computer science, operations research, and sociology. They could have implications for developing computer algorithms, aircraft scheduling, and species movement between regions.

Metric dimension of a graph is a well-studied concept. Recently, adjacency metric dimension of graph has been introduced. A set Qa⊂VG is considered to be an adjacency metric generator for G if u1,u2∈V\Qa (supposing each pair); there must exist a vertex q∈Qa along with the condition that q is indeed adjacent to one of u1,u2. The minimum number of elements in adjacency metric generator is the adjacency metric dimension of G, denoted by dimaG. In this work, we compute exact values of the adjacency metric dimension of circulant graph Cn1,2, Möbius ladder, hexagonal Möbius ladder, and the ladder graph.

Abstract Objectives The objective of the study is to measure the efficacy of ionic-iodine polymer complex [1] for clinical and radiological improvement in coronavirus disease 2019 (COVID-19) patients. Trial design The trial will be closed label, randomized and placebo-controlled with a 1:1:1:1 allocation ratio and superiority framework. Participants All PCR confirmed COVID-19 adult patients including non-pregnant females, with mild to moderate disease, will be enrolled from Shaikh Zayed Post-Graduate Medical Complex, Ali Clinic and Doctors Lounge in Lahore (Pakistan). Patients with any pre-existing chronic illness will be excluded from the study. Intervention and comparator In this multi-armed study ionic-iodine polymer complex with 200 mg of elemental iodine will be given using three formulations to evaluate efficacy. Patients will be receiving either encapsulated iodine complex of 200 mg (arm A), iodine complex syrup form 40 ml (arm B), iodine complex throat spray of 2 puffs (arm C) or empty capsule (arm D) as placebo; all three times a day. All the 4 arms will be receiving standard care as per version 3.0 of the clinical management guidelines for COVID-19 established by the Ministry of National Health Services of Pakistan. Main outcomes Primary outcomes will be viral clearance with radiological and clinical improvement. SARS-CoV-2 RT-PCR and HRCT chest scans will be done on the admission day and then after every fourth day for 12 days or till the symptoms are resolved. RT-PCR will only be shown as positive or negative while HRCT chest scoring will be done depending on the area and severity of lung involvement [2]. Time taken for the alleviation of symptoms will be calculated by the number of days the patient remained symptomatic. 30-day mortality will be considered as a secondary outcome. Randomisation Stratification for initial COVID-19 status (or days from initial symptoms as a proxy), age groups, gender, baseline severity of symptoms and co-morbidities will be used to ensure that the study arms remain balanced in size for the 1:1:1:1 allocation ratio. Randomization will be done using the lottery method. As patients are being admitted at different times, they will be recruited after obtaining their voluntary written informed consent following all standard protocols of the infection, control and disinfection. Blinding (masking) This is a quadruple (participants, care providers, investigators and outcomes assessors) blinded study where only the study’s Primary Investigator will have information about the arms and their interventions. Numbers to be randomised (sample size) 200 patients will be randomized into four groups with three experimental and one placebo arm. Trial Status Protocol Version Number is 2.3 and it is approved from IRB Shaikh Zayed Hospital with ID SZMC/IRB/Internal0056/2020 on July 14th, 2020. The recruitment is in progress. It was started on July 30, 2020, and the estimated end date for the trial is August 15, 2021. Trial registration Clinical Trial has been retrospectively registered on www.clinicaltrials.gov with registration ID NCT04473261 dated July 16, 2020. Full protocol The full protocol is attached as an additional file, accessible from the Trials website (Additional file 1). With the intention of expediting dissemination of this trial, the conventional formatting has been eliminated; this Letter serves as a summary of the key elements of the full protocol. The study protocol has been reported in accordance with the Standard Protocol Items: Recommendations for Clinical Interventional Trials (SPIRIT) guidelines.

Sphaeranthus indicus L. is a medicinal herb having widespread traditional uses for treating common ailments. The present research work aims to explore the in-depth phytochemical composition and in vitro reactivity of six different polarity solvents (methanol, n-hexane, benzene, chloroform, ethyl acetate, and n-butanol) extracts/fractions of S. indicus flowers. The phytochemical composition was accomplished by determining total bioactive contents, HPLC-PDA polyphenolic quantification, and UHPLC-MS secondary metabolomics. The reactivity of the phenolic compounds was tested through the following biochemical assays: antioxidant (DPPH, ABTS, FRAP, CUPRAC, phosphomolybdenum, and metal chelation) and enzyme inhibition (AChE, BChE, α-glucosidase, α-amylase, urease, and tyrosinase) assays were performed. The methanol extract showed the highest values for phenolic (94.07 mg GAE/g extract) and flavonoid (78.7 mg QE/g extract) contents and was also the most active for α-glucosidase inhibition as well as radical scavenging and reducing power potential. HPLC-PDA analysis quantified rutin, naringenin, chlorogenic acid, 3-hydroxybenzoic acid, gallic acid, and epicatechin in a significant amount. UHPLC-MS analysis of methanol and ethyl acetate extracts revealed the presence of well-known phytocompounds; most of these were phenolic, flavonoid, and glycoside derivatives. The ethyl acetate fraction exhibited the highest inhibition against tyrosinase and urease, while the n-hexane fraction was most active for α-amylase. Moreover, principal component analysis highlighted the positive correlation between bioactive compounds and the tested extracts. Overall, S. indicus flower extracts were found to contain important phytochemicals, hence could be further explored to discover novel bioactive compounds that could be a valid starting point for future pharmaceutical and nutraceuticals applications.

Let $\Omega $ be a connected graph and for a given $l$ -ordered partition of vertices of a connected graph $\Omega $ is represented as $\beta =\{\beta _{1},\beta _{2}, {\dots },\beta _{l}\}$ . The representation of a vertex $\mu \in V(\Omega)$ is the vector $r(\mu |\beta)=(d(\mu,\beta _{1}),d(\mu,\beta _{2}), {\dots }, d(\mu,\beta _{l}))$ . The partition $\beta $ is a resolving partition for vertices of $\Omega $ if all vertices of $\Omega $ having the unique representation with respect to $\beta $ . The minimum number of $l$ in the resolving partition for $\Omega $ is known as the partition dimension of $\Omega $ and represented as $pd(\Omega)$ . Resolving partition and partition dimension have multipurpose applications in networking, optimization, computer, mastermind games and modeling of chemical structures. The problem of computing constant values of partition dimension is NP-hard so one can find sharp bound for the partition dimension of graph. In this article, we computed the upper bound for the convex polytopes $\mathbb {E}_{n},~\mathbb {S}_{n},~\mathbb {T}_{n},~\mathbb {G}_{n},~\mathbb {Q}_{n}$ and flower graph $f_{n\times 3}$ .

ABSTRACT This molecular epidemiological study was designed to determine the antimalarial drug resistance pattern, and the genetic diversity of malaria isolates collected from a war-altered Federally Administered Tribal Area (FATA), in Pakistan. Clinical isolates were collected from Bajaur, Mohmand, Khyber, Orakzai and Kurram agencies of FATA region between May 2017 and May 2018, and they underwent DNA extraction and amplification. The investigation of gene polymorphisms in drug resistance genes (dhfr, dhps, crt, and mdr1) of Plasmodium falciparum and Plasmodium vivax was carried out by pyrosequencing and Sanger sequencing, respectively. Out of 679 PCR-confirmed malaria samples, 523 (77%) were P. vivax, 121 (18%) P. falciparum, and 35 (5%) had mixed-species infections. All P. falciparum isolates had pfdhfr double mutants (C59R+S108N), while pfdhfr/pfdhps triple mutants (C59R+S108N+A437G) were detected in 11.5% of the samples. About 97.4% of P. falciparum isolates contained pfcrt K76T mutation, while pfmdr1 N86Y and Y184F mutations were present in 18.2% and 10.2% of the samples. P. vivax pvdhfr S58R mutation was present in 24.9% of isolates and the S117N mutation in 36.2%, while no mutation in the pvdhps gene was found. Pvmdr1 F1076L mutation was found in nearly all samples, as it was observed in 98.9% of isolates. No significant anti-folate and chloroquine resistance was observed in P. vivax; however, mutations associated with antifolate-resistance were found, and the chloroquine-resistant gene has been observed in 100% of P. falciparum isolates. Chloroquine and sulphadoxine-pyrimethamine resistance were found to be high in P. falciparum and low in P. vivax. Chloroquine could still be used for P. vivax infection but need to be tested in vivo, whereas a replacement of the artemisinin combination therapy for P. falciparum appears to be justified.

Metal–organic structures/Networks (MONs) are useful in modern chemistry. It is common to use as a storehouse for the storage of gases, it plays a role in the separation of gases and purification. The most important feature of MONs is that it acts as a predecessor for the development of a large number of nanostructures. Furthermore, MONs reflect very useful chemical-physical properties, changing organic ligands, exchanging of ions, etc. The method used to forecast the natural behaviors among the chemical-physical specifications of the chemical compounds in their primitive network is known as topological indices or TIs. This numerical quantity is used in the method of forecasting. TIs of MONs shows a key role in the environmental and theoretical pharmacology and chemistry. Line graphs also have powerful applications in chemistry and predicting the boiling point of cycloalkanes. In this paper, we study Randić, atom bound connectivity, geometric arithmetic, Zagreb, Multiplicative Zagreb, redefined Zagreb indices and Zagreb coindices for line graph of first organic network L(MON1(χ)) and second organic network L(MON2(χ)),χ⩾2.

Polyphenyl is used in a variety of applications including high-vacuum devices, optics, and electronics, and in high-temperature and radiation-resistant fluids and greases, it has low volatility, ionizing radiation stability, and high thermal-oxidative properties. The structure of polyphenyls can be represented using a molecular graph, where atoms represent vertices and bonds between atom edges. In a chemical structure, an item/vertex v resolves two items v1 and v2 if dv1,v≠dv2,v; similarly, the ordered subset ϕ of vertices resolves each pair of distinct vertices named as the resolving set, and its minimum cardinality is described as metric dimension. In the pharmaceutical industry, the competition to find new chemical entities for treating a disease dictates larger project teams that encompass more extensive and diverse synthetic efforts directed at increasingly complicated activity spectra. In this paper, we determine the metric dimension of para-, meta-, and ortho-polyphenyl structures, which are used for structure-activity analysis of these polyphenyl structures.

Let G=VG,EG be the connected graph. For any vertex i∈VG and a subset B⊆VG, the distance between i and B is di;B=mindi,j|j∈B. The ordered k-partition of VG is Π=B1,B2,…,Bk. The representation of vertex i with respect to Π is the k-vector, that is, ri|Π=di,B1,di,B2,…,di,Bk. The partition Π is called the resolving (distinguishing) partition if ri|Π≠rj|Π, for all distinct i,j∈VG. The minimum cardinality of the resolving partition is called the partition dimension, denoted as pdG. In this paper, we consider the upper bound for the partition dimension of the generalized Petersen graph in terms of the cardinalities of its partite sets.

In computer networks, vertices represent hosts or servers, and edges represent as the connecting medium between them. In localization, some special vertices (resolving sets) are selected to locate the position of all vertices in a computer network. If an arbitrary vertex stopped working and selected vertices still remain the resolving set, then the chosen set is called as the fault-tolerant resolving set. The least number of vertices in such resolving sets is called the fault-tolerant metric dimension of the network. Because of the variety of applications of the metric dimension in different areas of sciences, many generalizations were proposed, and fault tolerant is one of them. In this paper, we computed the fault-tolerant metric dimension of triangular snake, ladder, Mobius ladder, and hexagonal ladder networks. It is important to observe that, in all these classes of networks, the fault-tolerant metric dimension and metric dimension differ by 1.

The position of a moving point in a connected graph can be identified by computing the distance from the point to a set of sonar stations which have been appropriately situated in the graph. Let Q = {q1, q2, ... , qk} be an ordered set of vertices of a graph G and a is any vertex in G, then the code/representation of a w.r.t Q is the k-tuple (r(a, q1), r(a, q2), ... , r(a, qk)), denoted by r(a|Q). If the different vertices of G have the different representations w.r.t Q, then Q is known as a resolving set/locating set. A resolving/locating set having the least number of vertices is the basis for G and the number of vertices in the basis is called metric dimension of G and it is represented as dim(G). In this paper, the metric dimension of Toeplitz graphs generated by two and three parameters denoted by Tn〈1, t〉 and Tn〈1, 2, t〉, respectively is discussed and proved that it is constant.

Abstract Background In Pakistan, artesunate (AS) in combination with sulfadoxine–pyrimethamine (SP) is the recommended treatment for uncomplicated Plasmodium falciparum malaria. Monitoring molecular markers of anti-malarial drug resistance is crucial for early detection and containment of parasite resistance to treatment. Currently, no data are available on molecular markers of artemisinin resistance (K13 mutations) in P. falciparum isolates from Pakistan. In this study, the prevalence of mutations associated with SP and artemisinin resistance was estimated in different regions of Pakistan. Methods A total of 845 blood samples that were positive for malaria parasites by microscopy or rapid diagnostic test were collected from January 2016 to February 2017 from 16 different sites in Pakistan. Of these samples, 300 were positive for P. falciparum by PCR. Polymorphisms in the P. falciparum dihydrofolate reductase (pfdhfr) and dihydropteroate synthase (pfdhps) genes were identified by pyrosequencing while polymorphisms in the propeller domain of the pfk13 gene were identified by Sanger sequencing. Results The prevalence of the PfDHFR 108N and 59R mutations was 100% and 98.8%, respectively, while the prevalence of PfDHFR 50R and 51I mutations was 8.6%. No mutation was observed at PfDHFR position 164. In PfDHPS, the prevalence of mutations at positions 436, 437, and 613 was 9.9%, 45.2%, and 0.4%, respectively. No mutations were found at PfDHPS positions 540 and 581. The prevalence of double PfDHFR mutants (59R + 108N) ranged from 93.8% to 100%, while the prevalence of parasites having the PfDHFR 59R + 108N mutations in addition to the PfDHPS 437G mutation ranged from 9.5% to 83.3% across different regions of Pakistan. Nine non-synonymous and four synonymous mutations were observed in the PfK13 propeller domain, none of which correspond to mutations validated to contribute to artemisinin resistance. Conclusion The absence of the highly resistant PfDHFR/PfDHPS quintuple mutant parasites and the lack of PfK13 mutations associated with artemisinin resistance is consistent with AS + SP being effective in Pakistan.

A spanning path in a graph G is called a Hamiltonian path. To determine which graphs possess such paths is an NP-complete problem. A graph G is called Hamiltonian-connected if any two vertices of G are connected by a Hamiltonian path. We consider here the family of Toeplitz graphs. About them, it is known only for n=3 that Tnp,q is Hamiltonian-connected, while some particular cases of Tnp,q,r for p=1 and q=2,3,4 have also been investigated regarding Hamiltonian connectedness. Here, we prove that the nonbipartite Toeplitz graph Tn1,q,r is Hamiltonian-connected for all 1

The efficiency of PV systems can be improved by accurate estimation of PV parameters. Parameter estimation of PV cells and modules is a challenging task as it requires accurate operation of PV cells and modules followed by an optimization tool that estimates their associated parameters. Mostly, population-based optimization tools are utilized for PV parameter estimation problems due to their computational intelligent behavior. However, most of them suffer from premature convergence problems, high computational burden, and often fall into local optimum solution. To mitigate these limitations, this paper presents an improved variant of particle swarm optimization (PSO) aiming to reduce shortcomings offered by conventional PSO for estimation of PV parameters. PSO is improved by introducing two strategies to control inertia weight and acceleration coefficients. At first, a sine chaotic inertia weight strategy is employed to attain an appropriate balance between local and global search. Afterward, a tangent chaotic strategy is utilized to guide acceleration coefficients in search of an optimal solution. The proposed algorithm is utilized to estimate the parameters of the PWP201 PV module, RTC France solar cell, and a JKM330P-72 PV module-based practical system. The obtained results indicate that the proposed technique avoids premature convergence and local optima stagnation of conventional PSO. Moreover, a comparison of obtained results with techniques available in the literature proves that the proposed methodology is an efficient, effective, and optimal tool to estimate PV modules and cells’ parameters.

Parameters associated with electrical equivalent models of the photovoltaic (PV) system play a significant role in the performance enhancement of the PV system. However, the accurate estimation of these parameters signifies a challenging task due to the higher computational complexities and non-linear characteristics of the PV modules/panels. Hence, an effective, dynamic, and efficient optimization technique is required to estimate the parameters associated with PV models. This paper proposes a double exponential function-based dynamic inertia weight (DEDIW) strategy for the optimal parameter estimation of the PV cell and module that maintains an appropriate balance between the exploitation and exploration phases to mitigate the premature convergence problem of conventional particle swarm optimization (PSO). The proposed approach (DEDIWPSO) is validated for three test systems; (1) RTC France solar cell, (2) Photo-watt (PWP 201) PV module, and (3) a practical test system (JKM330P-72, 310 W polycrystalline PV module) which involve data collected under real environmental conditions for both single- and double-diode models. Results illustrate that the parameters obtained from proposed technique are better than those from the conventional PSO and various other techniques presented in the literature. Additionally, a comparison of the statistical results reveals that the proposed methodology is highly accurate, reliable, and efficient.

In this work, we study the degree-based topological invariants, and the general sum-connectivity, A B C 4 , G A 5 , general Zagreb, G A , generalized Randić, and A B C indices of the line graphs of some rooted product graphs ( C n { P k } and C n { S m + 1 } ) are determined by menas of the concept of subdivision. Moreover, we also computed all these indices of the line graphs of the subdivision graphs of i-th vertex rooted product graph C i , r { P k + 1 } .

A topological index is a number related to the atomic index that allows quantitative structure–action/property/toxicity connections. All the more vital topological indices correspond to certain physico-concoction properties like breaking point, solidness, strain vitality, and so forth, of synthetic mixes. The idea of the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials was set up in the substance diagram hypothesis in light of vertex degrees. These indices are valuable in the investigation of calming exercises of certain compound systems. In this paper, we computed the first and second Zagreb index, the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials of the line graph of wheel and ladder graphs by utilizing the idea of subdivision.