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Consider a tree graph $ G $ with edge set $ E(G) $. The notation $ d_G(x) $ represents the degree of vertex $ x $ in $ G $. Let $ \mathfrak{f} $ be a symmetric real-valued function defined on the Cartesian square of the set of all distinct elements of the degree sequence of $ G $. A graphical edge-weight-function index for the graph $ G $, denoted by $ \mathcal{I}_\mathfrak{f}(G) $, is defined as $ \mathcal{I}_\mathfrak{f}(G) = \sum_{st \in E(G)} \mathfrak{f}(d_G(s), d_G(t)) $. This paper establishes the best possible bounds for $ \mathcal{I}_\mathfrak{f}(G) $ in terms of the order of $ G $ and parameter $ \mathfrak{p} $, subject to specific conditions on $ \mathfrak{f} $. Here, $ \mathfrak{p} $ can be one of the following three graph parameters: (ⅰ) matching number, (ⅱ) the count of pendent vertices, and (ⅲ) maximum degree. We also characterize all tree graphs that achieve these bounds. The constraints considered for $ \mathfrak{f} $ are satisfied by several well-known indices. We specifically illustrate our findings by applying them to the recently introduced Euler-Sombor index.

Doping of metal oxide with other suitable component can enhance its structural and characteristic properties. At the end it improves its overall application performances. This study introduces the novel synthesis of Al2O3@TiFe2O4 nanocomposite via co-precipitation for Cr(VI) removal. Unlike previous studies, this unique material exhibits superior adsorption capacity and efficiency. The comparative analysis with existing adsorbents underscores its potential for industrial effluent treatment. The nanocomposite’s recyclability and minimal efficiency loss over multiple cycles highlight its sustainability for wastewater treatment. Overall, this research fills existing gaps by presenting an advanced, environmentally friendly adsorbent, significantly enhancing the field of wastewater treatment and environmental remediation. The nanocomposite was characterized and used to study Cr(VI) adsorption characteristics. Amount, contact time, pH, and Cr(VI) concentration were found to influence the adsorption characteristics of the nanomaterial. The operating conditions of an adsorbent Al2O3@TiFe2O4 are optimized utilizing the analysis of the adsorption kinetic data. As shown by the R2 responses, which were close to unity, the adsorption behaviour of the nanomaterial is in excellent harmony with the linearized Langmuir isotherm with a maximum capacity of 312.52 mg/g. Additionally, Al2O3@TiFe2O4 active sites were more observed to be favorable for Cr(VI) ions at all experimental temperatures range, according to calculations of monolayer adsorption capacities using the Langmuir equation. The value of n was 0.40, 0.42, and 0.43, respectively 25, 35, and 45 °C. The adsorbent shows an adsorption capacity of 93 % with the RSD less than 2 %.

The co-doping of vanadium pentoxide (V2O5) with rare-earth (RE) elements, namely 1.5 % holmium (Ho) and 1.5 % ytterbium (Yb) has been conducted using an eco-friendly, straightforward hydrothermal approach to assess the combined effects on structural, optical, and photocatalytic properties. The application of the density functional theory (DFT) approach effectively examined the impact of RE ions on the photocatalytic efficiency of co-doped V2O5. The stable orthorhombic crystal structure of co-doped V2O5 has been confirmed using DFT and X-ray diffraction without a secondary phase. It appears that homogeneous nucleation occurs while heterogeneous nucleation slows down in co-doped samples, as evidenced by the larger crystallite sizes in co-doped samples compared to doped ones. It means a result, the co-doped samples exhibit photodegrades more quickly and have a higher rate constant than the doped samples. This is because they have less dislocation density (4.26 × 10−3 nm−2) and internal micro-strain (4.93 × 10−3). The bandgap and degradation efficiency are determined by the UV–vis spectroscopy and found to be 2.33 eV and 95 %, respectively, at the optimal pH of 7 in the visible range. The co-doped sample has a rate constant of 24 × 10−3 min−1, which is the highest in the RE-doped V2O5 system. This is a good reason to think of co-doped V2O5 as a possible catalyst.

This study explores green-synthesized delafossite CuAlO2 as a hole transport layer (HTL) in a CH3NH3SnI3-based perovskite solar cells (PSCs) with an FTO/CuAlO2/CH3NH3SnI3/WO3/Au structure. The performances of the cell have been theoretically investigated using SCAPS-1D. Interface defects significantly impact cell efficiency, as defect density increases from 1014 cm−3 to 1020 cm−3 efficiency reducing from 25.3% to 24.45% in the HTL/perovskite junction and from 25.2% to 17.8% in the perovskite/electron transport layer (ETL) interface. PCBM as buffer layer at perovskite/ETL interface compensates for power conversion efficiency (PCE) losses. Optimizing parameters reveals the delafossite CuAlO2-based, lead-free perovskite solar cell achieving peak efficiency at 26.74%, with VOC, JSC, and FF values of 0.99 V, 33.43 mA cm−2, and 81.05%, respectively. This research underscores delafossite CuAlO2 as a promising HTL for eco-friendly, stable CH3NH3SnI3-based perovskite solar cells, emphasizing its potential in enhancing device performance.

Many existing studies show that there exists a strong relationship between structures and characteristics of molecules. Topological indices are often used in modeling the properties of chemical compounds and biological activities in theoretical chemistry. Topological indices are numerical values associated with structures of molecules in such a way that they remain constant under graph isomorphism. Multiplicative Zagreb indices are among the famous topological indices that have been explored by numerous researchers in the last few years. The first objective of the present paper is to examine the importance of general multiplicative Zagreb indices for forecasting the enthalpy of formation of hydrocarbons using a data set of 25 benzenoid hydrocarbons. The second objective of this paper is to study molecular trees with a given order and with a given number of branching vertices or segments using general multiplicative (first and second) Zagreb indices. Sharp lower/upper bounds on these Zagreb indices for the aforementioned molecular trees are obtained and the graphs attaining these bounds are determined. Bounds on the classical multiplicative Zagreb and Narumi-Katayama indices are corollaries of the obtained results.

Organoclays have been proposed as efficient removal agents for colored wastewater treatment. In this study, organo-acid-activated clays were investigated for their ability to remove eosin Y dye molecules. Firstly, the clay was acid activated using sulfuric solution at 90 °C for overnight. Secondly, the resulting materials were treated by hexadecyltetramethylammonium bromide solutions to obtain organo-acid-activated clays. Several techniques were used, such as X-ray diffraction, carbon hydrogen nitrogen analysis, silicon-29 and carbon-13 solid-state nuclear magnetic resonance, and nitrogen adsorption isotherms. The cation exchange capacity values were reduced and the specific surface area values increased from 80.1 m2/g to 183.2 m2/g during the acid activation process. The resulting organo-acid-activated clays had a similar expansion of interlayer spacing of 3.90 nm with less intercalated hexadecyltetramethylammonium surfactants, ranging from 0.80 mmol/g to 1.22 mmol/g; 13C solid NMR indicated that the intercalated surfactants exhibited different degree of conformation. Many factors, including the solid dose, solution pH, amount of intercalated surfactants, and starting eosin-Y concentrations, were studied in relation to the removal capabilities of organo acid-activated clays. Based on the Langmuir model, the removal capacity of the organo acid-activated clays ranged from a minimum of 43.5 mg/g to a maximum of 79.3 mg/g, dependent on the surfactant quantities and acid activation degree. , and the pH. The removal percentage of eosin-Y was increased from 50.5 % to 80.8 % by treating the organo acid-activated clay with HCl solution before the removal procedure. Regeneration and reuse of two selected organo acid-activated clays were carried out for seven successive cycles, and a reduction in the initial efficiency was in the range of 26.4 %–30.1 %. However, for organoclay (without acid activation), approximately 52.1 % efficiency was maintained. Using the Langmuir model and mass balance equations, a single-stage adsorber design was suggested for different dye volumes at a constant starting concentration.

Despite the success of antibiotics in medicine, the treatment of bacterial infection is still challenging due to emerging resistance and suitable drug delivery system, therefore, innovative approaches focused on nanoparticles based antimicrobial drug delivery systems are highly desired. This research aimed to synthesize Cymbopogon citratus (C. citratus) aqueous extract-mediated copper oxide (CuO-Nps) conjugated with levofloxacin (LFX). The synthesized CuO NPs-LFX nano conjugate was confirmed by analysis using scanning electron microscopy (SEM), thermal gravimetric analysis (TGA), and infrared and ultraviolet/visible spectroscopy. Antibacterial activities were assessed in vitro through the agar well diffusion method against six bacterial strains of clinical relevance. CuO NPs confirmed by UV–Vis analysis absorption peak observed at 380 nm. TGA analysis showed 8.98% weight loss between the 400–800 °C temperature range. The functional group's presence was confirmed by FTIR analysis. Spherical shape nanoparticles with an average particle size of 55 nm were recorded by FESEM. Results from agar well diffusion assay showed that CuO NPs-LFX prohibited the development of both gram-positive and gram-negative bacteria at all established concentrations, and the antibacterial propensity was more pronounced as compared to bare CuO NPs, Levofloxacin and C. citratus aqueous extract alone. The results showed that gram-negative bacteria are more susceptible to CuO NPs-LFX nano conjugate and at 10 μgmL−1 concentration, form a 10.1 mm zone of inhibition (ZOI), whereas gram-positive bacteria on the same concentration form 9.5 mm ZOI. LFX-loaded CuO NPs antibacterial activity was observed higher than plant extract, bare CuO NPs, and standard drug (Levofloxacin). This study provides a novel approach for the fabrication of biogenic CuO NPs with antibacterial drug levofloxacin and their usage as nano antibiotic carriers against pathogenic bacteria, especially antibiotic-resistant microbes.

Bee products, abundant in bioactive ingredients, have been utilized in both traditional and contemporary medicine. Their antioxidant, antimicrobial, and anti-inflammatory properties make them valuable for food, preservation, and cosmetics applications. Honeybees are a vast reservoir of potentially beneficial products such as honey, bee pollen, bee bread, beeswax, bee venom, and royal jelly. These products are rich in metabolites vital to human health, including proteins, amino acids, peptides, enzymes, sugars, vitamins, polyphenols, flavonoids, and minerals. The advancement of nanotechnology has led to a continuous search for new natural sources that can facilitate the easy, low-cost, and eco-friendly synthesis of nanomaterials. Nanoparticles (NPs) are actively synthesized using honeybee products, which serve dual purposes in preventive and interceptive treatment strategies due to their richness in essential metabolites. This review aims to highlight the potential role of bee products in this line and their applications as catalysts and food preservatives and to point out their anticancer, antibacterial, antifungal, and antioxidant underlying impacts. The research used several online databases, namely Google Scholar, Science Direct, and Sci Finder. The overall findings suggest that these bee-derived substances exhibit remarkable properties, making them promising candidates for the economical and eco-friendly production of NPs.

This paper investigates the global asymptotic stability problem for a class of Takagi-Sugeno fuzzy Clifford-valued delayed neural networks with impulsive effects and leakage delays using the system decomposition method. By applying Takagi-Sugeno fuzzy theory, we first consider a general form of Takagi-Sugeno fuzzy Clifford-valued delayed neural networks. Then, we decompose the considered -dimensional Clifford-valued systems into -dimensional real-valued systems in order to avoid the inconvenience caused by the non-commutativity of the multiplication of Clifford numbers. By using Lyapunov-Krasovskii functionals and integral inequalities, we derive new sufficient criteria to guarantee the global asymptotic stability for the considered neural networks. Further, the results of this paper are presented in terms of real-valued linear matrix inequalities, which can be directly solved using the MATLAB LMI toolbox. Finally, a numerical example is provided with their simulations to demonstrate the validity of the theoretical analysis.

Various strategies have been adapted to fabricate stable organic-inorganic hybrid perovskite (PVT) solar cells (PSCs). The triple-cation (CH3NH3+ (MA+), CH3(NH2)2+ (FA+), and Cs+) along with dual-anion (I− and Br−)-based PVT (TC-PVT) layer offers better stability than single cation-based PVTs. The deprivation of the PVT absorber is also influenced by the interface of the absorber with the charge transport layer (electron transport layer (ETL) and hole transport layer (HTL)). Here, the degradation of the TC-PVT coated on Al-doped zinc oxide (AZO) as well as FTO/AZO/TC-PVT/HTL structured PSC was examined for various Al to Zn molar ratio (RAl/Zn) of AZO. The PL decay study of FTO/AZO/TC-PVT revealed that the lowest degradation in the power (35.38%) was observed for the AZO with RAl/Zn of 5%. Furthermore, the PV cell parameters of the PSCs were analytically determined to explore the losses in the PSCs during degradation. The shunt resistance reduction was maximum (50.32%) for RAl/Zn = 10%, whereas, minimum shunt loss (7.33%) for RAl/Zn of 2%. The highest loss due to series resistance was observed for RAl/Zn of 0%. The changes in diode ideality factor (n) and reverse saturation current density (J0) were the smallest for RAl/Znof 10%.

Na-magadiite materials were prepared from a gel containing a silica source, sodium hydroxide, and water via hydrothermal treatment at different temperatures (130 °C to 170 °C) and periods of time (1 day to 10 days). In this study, four silica sources were selected (fumed silica, colloidal silica, Ludox HS-40%, and Ludox AS-40%). Variable conditions such as sodium hydroxide and water contents were explored at a specific temperature and reaction time. The obtained materials were characterized by using X-ray diffraction (XRD), thermogravimetry differential thermal analysis TG-DTA, scanning electron microscopy with energy dispersive X-ray spectroscopy (SEM-EDX), Fourier Transform Infrared spectroscopy (FTIR), solid 29Si magic angle spinning magnetic nuclear resonance (MAS MNR, and nitrogen adsorption isotherms. A pure Na-magadiite phase was obtained from the four silica sources at a synthesis temperature of 150 °C after a period of one to two days with a characteristic basal spacing of 1.54 nm. At a longer reaction time of 3 days and a higher temperature of 170 °C, Na-kenyaite with a basal spacing of 2.01 nm was achieved, in addition to a quartz phase. The content of water or sodium hydroxide in the gel affected the nature of the prepared phases. A cauliflower-like morphology was obtained from colloidal silica sources, while a different morphology was achieved using solid fumed silica. The 29Si solid NMR confirmed the presence of Q3 and Q4 silicon sites in the Na-magadiite materials. The optimal Na-magadiite materials at 150 °C for 2 days were assessed for their ability to remove Basic Blue-41 dye from artificially contaminated aqueous solution. The Langmuir equation was used to estimate the maximum removal capacity. A maximum removal capacity of 219 mg/g was achieved using Na-magadiite prepared from a Ludox-HS40% silica source, and a maximum removal capacity of 167 mg/g was observed for Na-magadiite prepared from fumed silica. Basic Blue-4’s removal percentage was enhanced at basic pH levels (8 to 10) to a maximum of 95%. These materials could be regenerated for seven cycles of reuse with a reduction of 27 to 40% of the original values. Therefore, Na-magadiite materials are promising and efficient removal agents for the removal of Basic Blue-41 from effluents.

It is well known that crisp graph theory is saturated. However, saturation in a fuzzy environment has only lately been created and extensively researched. It is necessary to consider m components for each node and edge in an m-polar fuzzy graph. Since there is only one component for this idea, we are unable to manage this kind of circumstance using the fuzzy model since we take into account m components for each node as well as edges. Again, since each edge or node only has two components, we are unable to apply a bipolar or intuitionistic fuzzy graph model. In contrast to other fuzzy models, mPFG models produce outcomes of fuzziness that are more effective. Additionally, we develop and analyze these kinds of mPFGs using examples and related theorems. Considering all those things together, we define saturation for a m-polar fuzzy graph (mPFG) with multiple membership values for both vertices and edges; thus, a novel approach is required. In this context, we present a novel method for defining saturation in mPFG involving m saturations for each element in the membership value array of a vertex. This explains α-saturation and β-saturation. We investigate intriguing properties such as α-vertex count and β-vertex count and establish upper bounds for particular instances of mPFGs. Using the concept of α-saturation and α-saturation, block and bridge of mPFG are characterized. To identify the α-saturation and β-saturation mPFGs, two algorithms are designed and, using these algorithms, the saturated mPFG is determined. The time complexity of these algorithms is O(|V|3), where |V| is the number of vertices of the given graph. In addition, we demonstrate a practical application where the concept of saturation in mPFG is applicable. In this application, an appropriate location is determined for the allocation of a facility point.

The nanostructures synthesized using the green chemistry method have recently attracted the attention of scientists due to their significance in many scientific domains. This work provides an overview of the biosynthesis of zinc oxide (ZnO) nanosheets (NSs) using Phyllanthus emblica plant (PEP) extract. X-ray diffraction analysis (XRD), X-ray photoelectron spectroscopy (XPS), scanning electron microscopy (SEM), and Fourier transform infrared (FTIR) were used to analyze the synthesized ZnO-NSs. Evaluation of the antibacterial activity of biosynthesized ZnO-NSs was performed. ZnO-NSs exhibit effective antibacterial activity against Gram-positive (S. pyogenes and S. aureus) and Gram-negative (S. typhi and E. coli) bacterial strains. S. typhi is the most sensitive microbe towards ZnO-NSs and formed a 21 mm zone of inhibition (ZOI). ZnO-NSs are also tested as a photocatalyst in the degradation of methyl orange (MO) and rhodamine B (RB). The degradation rate of MO was 90%, and RB was 96% after being exposed to UV light for 120 min. The as-synthesized ZnO-NSs exhibited selective dye degradation and showed relatively better photocatalytic activity for positively charged (cationic) dyes. This work could lead to the fabrication of high-yield photocatalysts, which have the potential to degrade textile dyes from aqueous solution.

Cutaneous melanoma is a severe and life-threatening form of skin cancer with growing incidences. While novel interventions have improved prognoses for these patients, early diagnosis of targeted treatment remains the most effective approach. MicroRNAs have grown to good use as potential biomarkers for early detection and as targets for treatment. miR-155 is well-studied for its role in tumor cell survival and proliferation in various tissues, although its role in melanoma remains controversial. In silico data analysis was performed in the dbDEMC v.3 to identify differentially expressed miRNA. We validated gene targets in melanoma using TarBase v8.0 and miRPath v3.0 and determined protein-protein interactions of the target genes. One hundred forty patients (age range 21–90 years) with cutaneous melanoma who underwent resection were included. Molecular assessment using Real-Time RT-qPCR, clinicopathological associations, and a literature review for the different roles of miR-155 in melanoma were performed. Analysis of the dbDEMC reveals controversial findings. While there is evidence of upregulation of miR-155 in primary and metastatic melanoma samples, others suggest decreased expression in later-stage melanoma and cases with brain metastasis. miR-155 has been overexpressed in prior cases of melanoma and precancerous lesions, and it was found to be dysregulated when compared to benign nevi. While miR-155 expression was associated with favorable outcomes in some studies, others showed an association with metastasis. Patients with high levels of miR-155 also noted reduction after receiving anti-PD-1 treatment, correlated with more prolonged overall survival. In our patient’s cohort, 22.9% relapsed during treatment, and 45% developed recurrence, associated with factors such as lymph node infiltration, high mitotic index, and positive staining for CD117. Although overall analysis revealed miR-155 downregulation in melanoma specimens compared to non-cancer tissues, increased expression of miR-155 was associated with cases of superficial spreading melanoma subtype (p = 0.005) and any melanoma with a high mitotic rate (p = 0.010). The analysis did not identify optimum cutoff values to predict relapse, recurrence, or mortality. In conclusion, miR-155 could have, in part, a potential prognostic utility in cutaneous melanoma. Further mechanistic studies are required to unravel the multifunctional role of miR-155 in melanoma.

The Ambartsumian equation in view of the q-calculus is investigated in this paper. This equation is of practical interest in the theory of surface brightness in the Milky Way. Two approaches are applied to obtain the closed form solution. The first approach implements a direct series assumption while the second approach is based on the Adomian decomposition method. The two approaches lead to a unique power series of arbitrary powers. Furthermore, the convergence of the obtained series is theoretically proven. In addition, we showed that the present solution reduces to the results in the relevant literature when the quantum calculus parameter tends to 1.

A connected graph in which no edge lies on more than one cycle is called a cactus graph (also known as Husimi tree). A bond incident degree (BID) index of a graph G is defined as ∑uv∈EGfdGu,dGv, where dGw denotes the degree of a vertex w of G, EG is the edge set of G, and f is a real-valued symmetric function. This study involves extremal results of cactus graphs concerning the following type of the BID indices: IfiG=∑uv∈EGfidGu/dGu+fidGv/dGv, where i∈1,2, f1 is a strictly convex function, and f2 is a strictly concave function. More precisely, graphs attaining the minimum and maximum Ifi values are studied in the class of all cactus graphs with a given number of vertices and cycles. The obtained results cover several well-known indices including the general zeroth-order Randić index, multiplicative first and second Zagreb indices, and variable sum exdeg index.

In this work, we introduce a new type of generalized mixed-type quadratic-additive functional equation and obtain its general solution. The main goal of this work is to investigate Ulam stability of this mixed type of quadratic-additive functional equation in the setting of non-Archimedean fuzzy φ-2-normed space and non-Archimedean Banach space using the direct and fixed point approaches by taking into our account two cases: even mapping and odd mapping.

Several research works have investigated the direct supply of renewable electricity to electrolysis, particularly from photovoltaic (PV) and wind generator (WG) systems. Hydrogen (H2) production based on solar energy is considered to be the newest solution for sustainable energy. Different technologies based on solar energy which allow hydrogen production are presented to study their benefits and inconveniences. The technology of water decomposition based on renewable energy sources, to produce hydrogen, can be achieved by different processes (photochemical systems; photocatalysis systems, photo-electrolysis systems, bio-photolysis systems, thermolysis systems, thermochemical cycles, steam electrolysis, hybrid processes, and concentrated solar energy systems). A comparison of the different methods for hydrogen production based on PV and WG systems was given in this study. A comparative study of different types of electrolyzers was also presented and discussed. Finally, an economic assessment of green hydrogen production is given. The hydrogen production cost depends on several factors, such as renewable energy sources, electrolysis type, weather conditions, installation cost, and the productivity of hydrogen per day. PV/H2 and wind/H2 systems are both suitable in remote and arid areas. Minimum maintenance is required, and a power cycle is not needed to produce electricity. The concentrated CSP/H2 system needs a power cycle. The hydrogen production cost is higher if using wind/H2 rather than PV/H2. The green energy sources are useful for multiple applications, such as hydrogen production, cooling systems, heating, and water desalination.

Hazardous dyes in industrial wastewater are an internationally recognized issue for community health. Nanoparticles synthesized through green protocols are a fascinating research field with numerous applications. The current study mainly aimed to investigate the degradation of Congo red (CR) dye under UV light in the presence of H2O2 and the photocatalytic activity of copper oxide nanoparticles (CuONPs). For CuONP formation, Citrus maxima extract contains a high number of phytochemical constituents. The size of CuONPs ranges between 25 and 90 nm. The photocatalytic activity of CuONPs with the addition of H2O2 was observed and analyzed under UV light to eliminate CR dye. The UV light caused the decomposition of H2O2, which produced ·OH radicals. The results revealed a significant increment in dye degradation during the presence of H2O2. The effect of concentration on the degradation of the CR dye was also studied. The degradation pathway of organic pollutants was reputable from the hydroxy radical medicated degradation of CR. Advanced Oxidation Treatment depends on the in situ production of reactive ·OH species and is presented as the most effective procedure for decontamination. The biological activity of CuONPs was evaluated against Escherichia coli Bacillus subtillis, Staphylococcus aureus, Shigella flexenari, Acinetobacter Klebsiella pneumonia, Salmonella typhi and Micrococcus luteus. The newly synthesised nanomaterials showed strong inhibition activity against Escherichia coli (45%), Bacillus subtilis (42%) and Acinetobacter species (25%). The activity of CuONPs was also investigated against different fungus species such as: Aspergillus flavus, A. niger, Candida glabrata, T. longifusus, M. Canis, C. glabrata and showed a good inhibition zone against Candida glabrata 75%, Aspergillus flavus 68%, T. longifusus 60%. The materials showed good activity against C. glaberata, A. flavus and T. longifusus. Furthermore, CuONPs were tested for antioxidant properties using 2, 2 diphenyl-1-picrylhydrazyl) (DPPH).

Glauconite (Gl), a naturally occurring clay material, was utilized as an affordable and ecologically friendly adsorbent to explore its capturing capacity towards Congo red (CR) dye from textile industrial waste effluent. To improve adsorption and removal effectiveness, a modification technique utilizing polyaniline (PAN) was investigated. An X-ray diffractometer (XRD), a scanning electron microscope (SEM), and Fourier transformer infrared (FTI-R) were applied as strong familiar characterization techniques for all used adsorbents. The effects of starting concentration, contact duration, adsorbent dose, pH, and temperature on the adsorption process were also studied. The reusability of the adsorbent was studied over four adsorption cycles. The results show that PAN modification of Gl enhances the effectiveness of CR elimination. The clearance efficiency of raw and modified glauconite at 25 °C and pH 7 was 77% and 91%, respectively. The kinetics and isotherms of Congo red dye adsorption were investigated using batch studies to determine the impacts of various experimental conditions. The maximum adsorption capacity of the glauconite/polyaniline (Gl/PAN) nanocomposite rose from 11.9 mg/g for Gl to 14.1 mg/g in accordance with the isotherm analysis, which shows that the Langmuir isotherm properly characterizes the experimental data. The pseudo-second-order model (R2 = 0.998) properly expresses the experimental data. The reusability research proved that the adsorbents may be reused effectively. The overall results suggest that the modified Gl by PAN might be used as a low-cost, natural adsorbent for eliminating CR color from textile effluent.

Let A be a commutative ring with unity and let set of all zero divisors of A be denoted by ZA. An ideal ℐ of the ring A is said to be essential if it has a nonzero intersection with every nonzero ideal of A. It is denoted by ℐ≤eA. The generalized zero-divisor graph denoted by ΓgA is an undirected graph with vertex set ZA∗ (set of all nonzero zero-divisors of A) and two distinct vertices x1 and x2 are adjacent if and only if annx1+annx2≤eA. In this paper, first we characterized all the finite commutative rings A for which ΓgA is isomorphic to some well-known graphs. Then, we classify all the finite commutative rings A for which ΓgA is planar, outerplanar, or toroidal. Finally, we discuss about the domination number of ΓgA.

Let S be a commutative ring with unity, and a set of nonunit elements is denoted by WS. The coannihilator graph of S, denoted by AG′S, is an undirected graph with vertex set WS∗ (set of all nonzero nonunit elements of S), and α∼β is an edge of AG′S⇔α∉αβS or β∉αβS, where δS denotes the principal ideal generated by δ∈S. In this study, we first classify finite ring S, for which AG′S is isomorphic to some well-known graph. Then, we characterized the finite ring S, for which AG′S is toroidal or projective.

Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, and topological spaces. This provides sufficient motivation to researchers to review various concepts and results from the realm of abstract algebra in the broader framework of fuzzy setting. In this paper, we introduce the notions of int-soft m,n-ideals, int-soft m,0-ideals, and int-soft 0,n-ideals of semigroups by generalizing the concept of int-soft bi-ideals, int-soft right ideals, and int-soft left ideals in semigroups. In addition, some of the properties of int-soft m,n-ideal, int-soft m,0-ideal, and int-soft 0,n-ideal are studied. Also, characterizations of various types of semigroups such as m,n-regular semigroups, m,0-regular semigroups, and 0,n-regular semigroups in terms of their int-soft m,n-ideals, int-soft m,0-ideals, and int-soft 0,n-ideals are provided.

Natural genetic material may shed light on gene expression mechanisms and aid in the detection of genetic disorders. Single Nucleotide Polymorphism (SNP), small insertions and deletions (indels), and major chromosomal anomalies are all chromosomal abnormality-related disorders. As a result, several methods have been applied to analyze DNA sequences, which constitutes one of the most critical aspects of biological research. Thus, numerous mathematical and algorithmic contributions have been made to DNA analysis and computing. Cost minimization, deployment, and sensitivity analysis to many factors are all components of sequencing platforms built on a quantitative framework and their operating mechanisms. This study aims to investigate the role of DNA sequencing and its representation in the form of graphs in the analysis of different diseases by means of DNA sequencing.

The quantum calculus, q-calculus, is a relatively new branch in which the derivative of a real function can be calculated without limits. In this paper, the falling body problem in a resisting medium is revisited in view of the q-calculus to the first time. The q-differential equations describing the vertical velocity and distance of the body are obtained. Accordingly, exact expressions for the vertical velocity and the vertical distance are provided. The solutions are expressed in terms of the small q-exponential function which is an elementary function in the q-calculus. The dimensionality of the obtained formulae of the velocity and the distance are also analyzed. In addition, the present exact solutions reduce to the corresponding solutions in classical Newtonian mechanics when the quantum parameter q tends to one.

In this paper, we introduce and investigate an ideal-based dot total graph of commutative ring R with nonzero unity. We show that this graph is connected and has a small diameter of at most two. Furthermore, its vertex set is divided into three disjoint subsets of R. After that, connectivity, clique number, and girth have also been studied. Finally, we determine the cases when it is Eulerian, Hamiltonian, and contains a Eulerian trail.

In this paper, the authors investigate the Hyers–Ulam stability results of the quadratic functional equation in Banach spaces and non-Archimedean Banach spaces by utilizing two different techniques in terms of direct and fixed point techniques.

In this paper, we present a generalization of one of the theorems in Partitions with parts separated by parity introduced by George E. Andrews, and give its bijective proof. Further variations of related partition functions are studied resulting in a number of interesting identities.

Competition graph is a graph which constitutes from a directed graph (digraph) with an edge between two vertices if they have some common preys in the digraph. Moreover, Fuzzy competition graph (briefly, FCG) is the higher extension of the crisp competition graph by assigning fuzzy value to each vertex and edge. Also, Interval-valued FCG (briefly, IVFCG) is another higher extension of fuzzy competition graph by taking each fuzzy value as a sub-interval of the interval [ 0 , 1 ] . This graph arises in many real world systems; one of them is discussed as follows: Each and every species in nature basically needs ecological balance to survive. The existing species depends on one another for food. If there happens any extinction of any species, there must be a crisis of food among those species which depend on that extinct species. The height of food crisis among those species varies according to their ecological status, environment and encompassing atmosphere. So, the prey to prey relationship among the species cannot be assessed exactly. Therefore, the assessment of competition of species is vague or shadowy. Motivated from this idea, in this paper IVFCG is introduced and several properties of IVFCG and its two variants interval-valued fuzzy k-competition graphs (briefly, IVFKCG) and interval-valued fuzzy m-step competition graphs (briefly, IVFMCG) are presented. The work is helpful to assess the strength of competition among competitors in the field of competitive network system. Furthermore, homomorphic and isomorphic properties of IVFCG are also discussed. Finally, an appropriate application of IVFCG in the competition among the production companies in market is presented to highlight the relevance of IVFCG.

In this current work, we introduce the finite variable additive functional equation and we derive its solution. In fact, we investigate the Hyers–Ulam stability results for the finite variable additive functional equation in fuzzy normed space by two different approaches of direct and fixed point methods.

Conditions that are necessary for the relative annihilator in lower B C K -semilattices to be a prime ideal are discussed. Given the minimal prime decomposition of an ideal A, a condition for any prime ideal to be one of the minimal prime factors of A is provided. Homomorphic image and pre-image of the minimal prime decomposition of an ideal are considered. Using a semi-prime closure operation “ c l ”, we show that every minimal prime factor of a c l -closed ideal A is also c l -closed.