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In this study, the purpose was to make use of metaphors to determine the difficulties experienced by secondary school 6th and 7th grade students in the process of learning the subject of circle. In the study, the phenomenological research design, one of qualitative research methods, was used. The research data were collected from a total of 140 secondary school students (80 6th grade, 60 7th grade), who were asked to fill in the blanks in the statement of "While learning the geometric shape of circle, I experience the biggest difficulty in …. because …". For the analysis of the data, content analysis method was used. The results revealed 42 different valid metaphors. These metaphors were gathered under seven categories. Among these categories, the most frequently produced ones were "problems related to the shapes drawn in a circle and tangent problems with circles." Moreover, in the study, it was found that when compared to the 6th grade students, the 7th grade students produced more metaphors related to the difficulties experienced in the teaching process of circle. Based on all these results and the students' perceptions, suggestions were put forward regarding the development of students' positive perceptions and understanding of geometry teaching.

The present study aimed to examine an effective mathematics teacher's qualifications from the perspectives of secondary school students. The study was a qualitative one carried out with the descriptive model. The study was carried out with 40 students from all the four class grades of a secondary school (10 students from each class grade) in Turkey in the academic year of 2017-2018. The participants were determined using the purposeful sampling method. As the data collection tool, semi-structured interview questions were used, and the interviews were audio-recorded to avoid any data loss. For the analysis of the data, first, the interviews were transcribed and then analysed using the content analysis method. The results revealed that according to the students' views, the qualifications of an effective mathematics teacher were grouped under four categories: "personal traits", "communication skills", "class management and teaching methods" and "measurement-evaluation". It was also seen that regardless of the school type they would choose for their high school education, all the students had an idea about the qualifications of a mathematics teacher and that it was not much easy to become an effective mathematics teacher. In addition, it was found that there were differences between the views of the 5th, 6th, 7th and 8th grade students with respect to their cognitive levels and needs. In line with the qualifications examined, various suggestions were put forward to increase the effectiveness of the mathematics teachers and of the process of mathematics teaching.

Porous materials, for example, metalnatural structures (MOFs) and their discrete partners metalnatural polyhedra (MOPs), that are built from coordinatively unsaturated inorganic hubs show incredible potential for application in gas adsorption/partition cycles, catalysis, and arising openings in hardware, optics, detecting, and biotechnology. A well-known hetero-bimetallic metalorganic polyhedra of this discrete partners metalnatural polyhedra (MOPs) class is cuboctahedral bi-metallic stricture. In this paper, we discuss the stricture of Hetero-bimetallic metalorganic polyhedra (cuboctahedral bi-metallic). Also, we computed the topological indices based on the degree of atoms in this cuboctahedral bi-metallic structure.

Borophene nanosheets appear in various sizes and shapes, ranging from simple planar structures to complicated polyhedral formations. Due to their unique chemical, optical, and electrical properties, Borophene nanosheets are theoretically and practically attractive and because of their high thermal conductivity, boron nanosheets are suitable for efficient heat transmission applications. In this paper, temperature indices of borophene nanosheets are computed and these indices are employed in QSPR analysis of attributes like Young's modulus, Shear modulus, and Poisson's ratio of borophene nanosheets and borophene β12 sheets. The regression model for the F-Temperature index is discovered to be the best fit for shear modulus, the reciprocal product connectivity temperature index is discovered to be fit for Poisson's ratio and the second hyper temperature index is discovered to be fit for Young's modulus based on the correlation coefficient. [ABSTRACT FROM AUTHOR]

A variety of graphical invariants have been described and tested, offering lots of applications in the fields of nanochemistry, computational networks, and different scientific research areas. One commonly studied group of invariants is the topological index, which allows research on the chemical, biological, and physical properties of a chemical structure. Topological indices are numerical quantities that can be used to describe the properties of the molecular graph. In this article, we draw from the analytically closed formulas of molecular structures of polycyclic aromatic hydrocarbons by calculating temperature-based topological indices. Our results in this paper may be useful to better understand many physical and chemical properties of polycyclic aromatic hydrocarbons. [ABSTRACT FROM AUTHOR]

The phenylenes exhibit unique physicochemical properties due to their aromatic and antiaromatic rings. The main object of this paper is to determine the expected values of the arithmetic-geometric and augmented Zagreb indices for random phenylene chains. The comparisons between the expected values of these indices with respect to the random phenylene chains have been determined explicitly. The graphical illustrations have been given in terms of the differences between the expected values of these indices. [ABSTRACT FROM AUTHOR]

Graph theory plays a significant role in the applications of chemistry, pharmacy, communication, maps, and aeronautical fields. A benzenoid is a class of chemical compounds with at least one benzene ring(hexagon as a graph) and resonance bonds in the benzene ring give increased stability in benzenoids. The molecules of chemical compounds are modeled as a graph to study the properties of the compounds. The geometric structure of the compound relates to a few physical properties such as boiling point, enthalpy, π-electron energy, and molecular weight. Entropy is a thermodynamic function in physics that measures the randomness and disorder of molecules in a particular system or process based on the diversity of configurations that molecules might take. Degree-based entropy is used to address a wide range of problems in the domains of mathematics, biology, chemical graph theory, organic and inorganic chemistry, and other disciplines. This paper focusses on computing analytical expressions of degree-based entropy measures for benzenoid systems. [ABSTRACT FROM AUTHOR]

In this article, a quantitative structure-property relationship is performed for the prediction of six physico-chemical properties of 16 alkaloid structures using three different types of degree-based topological indices. Chemical structures are considered as graphs, where elements are taken as vertices and bounds between them are taken as edges. We defined two new degree-based topological indices, namely, the "modified harmonic index" and the "advanced harmonic index," to analyze and examine the properties of alkaloids. The topological indices and experimental values act as the inputs of linear and quadratic regression models. The correlation values and p-values for all the indices are significant which describe the validity and usefulness of the results. The outcomes determined in this article assist pharmacists and chemists in studying the structures of alkaloids for use in daily life, agriculture, pharmacy, and industries. The study of chemical structures with the help of molecular descriptors and regression models is a theoretical method that saves time and money. [ABSTRACT FROM AUTHOR]

The F-index is the whole of 3D squares of vertex degrees in a chart G. It as of late turned into the subject of a few investigates because of its extraordinary capability of uses. The point of this paper is to register the F-record of triangular prickly plant chain, square desert flora chains, 6-sided cactus restraints and polyomino restraints. In addition, we decided the extremal chains in the desert plant chains and polyomino fastens as for the F-index.

A large number of medical experiments have confirmed that the features of drugs have a close correlation with their molecular structure. Drug properties can be obtained by studying the molecular structure of corresponding drugs. The calculation of the topological index of a drug structure enables scientists to have a better understanding of the physical chemistry and biological characteristics of drugs. In this paper, we obtain the exact formulas by virtue of the edge-partition method for m S O α (G) and K A (α , β) 1 (G). By the appropriate choice of parameters α and β , several new/old inequalities that reveal relationships between various topological indices are obtained. Our results remedy the lack of medicine experiments, thus providing a theoretical basis for pharmaceutical engineering. [ABSTRACT FROM AUTHOR]

In Sri Lanka as well as the rest of the globe, cancer is the top cause of mortality. One of the key medicines in treating tumors is anticancer medications and delivery dendrimers. To prevent the formation of the rapid proliferation of cancer cells, several tests were carried out. Because of this, research on dendrimers and anti-cancer medications is crucial. Topological indices (TIs) are molecular descriptors numerical values corresponding to the physical characteristics of a molecule's chemical structure. It costs money to determine a molecule's physical characteristics in a lab since it takes a lot of materials, medications, and time. Therefore, the relevant information about molecules may be obtained by computing TIs. This study's goals are to compute hitherto uncalculated eccentricity-based TIs for various anticancer structures and to use curvilinear regression models to forecast the physical characteristics of particular anticancer medications. These anticancer medications were given different TIs developed in this work, allowing the researchers to understand the physical, physicochemical, and chemical characteristics related to them. In addition comparative study of the novel indices with some well-known and mostly used indices in structure–property modeling and anticancer drugs in performed. [ABSTRACT FROM AUTHOR]

Several graph features have been used to distinguish the construction of entropy-based measurements from the structure of chemical graphs and complicated networks. The graph entropy metric has piqued the interest of scientists due to its possible applicability in a variety of domains. In this paper, we look at the chemical graph of silicon–carbon SiC3-I and the crystallographic structure of silicon–carbon SiC3-II. This two-dimensional structure has the potential to revolutionize optoelectronic and electrical technologies as a direct wide band gap semiconducting material. More intriguingly, the semiconducting SiC3 sheet has a significant visible-light adsorption ability. SiC3 nanosheets have a wide range of electrical characteristics, making them ideal for nano-electronics and photo-voltaic. This advantageous molecular structure, silicon carbon, is examined. More preciously in this paper, we have computed the entropy measures for SiC3-I and SiC3-II based on the topologies indices. Some comparison work is offered in addition to the analytical analysis of the entropy measure of silicon–carbon. Bases on these numerical tables and corresponding graphs we conclude that as entropy measures are the better predicator than the topological indices of physio-chemical properties. [ABSTRACT FROM AUTHOR]

Edge irregular mapping or vertex mapping h : V (G) −→ {1, 2, 3, ..., s} is a mapping of vertices in such a way that all edges have distinct weights. We evaluate weight of any edge by using equation wth(cd) = h(c) + h(d), ∀c, d ∈ V (G) and ∀cd ∈ E(G). Edge irregularity strength denoted by es(G) is a minimum positive integer used to label vertices to form edge irregular labeling. In this paper, we find exact value of edge irregularity strength of linear phenylene graph PHn, Bn graph and different families of snake graph. [ABSTRACT FROM AUTHOR]

Polyphenyls and their derivatives, which are used in heat exchangers, drug synthesis, organic synthesis, and so on, have attracted the attention of researchers in various fields for many years. In this paper, we determine the expected values of the augmented Zagreb and arithmetic-geometric indices for this class of conjugated hydrocarbons. The comparisons between the expected values of these indices concerning the random polyphenyl chains have been determined explicitly. The graphical illustrations have been given in terms of the differences between the expected values of these indices. [ABSTRACT FROM AUTHOR]

The threat of developing a cancer therapy has been there for the past two to three decades. Almost 10 million people worldwide are affected by this illness each year. Anticancer medications are those that are used to treat cancer, a malignant condition. These anticancer medications come in a variety of types, such as hormones, antimetabolites, and alkalizing agents. Numerous studies demonstrate a close relationship between the chemical structure of anticancer drugs and alkane characteristics including boiling, melting, and enthalpy. Some antiviral medications that are thought to hold promise for the treatment of cancer are examined in the planned research. For the chemical graphs of these medications, ST-polynomials based on status distance are computed. Numerous topological indices (TI) that depend on the status distance may be constructed by applying these polynomials. Additionally, the TIs are used in the creation of the QSPR models to estimate some of the physicochemical characteristics of these medications. Curve fitting is used to produce the QSPR research. [ABSTRACT FROM AUTHOR]

Benzene is aromatic organic compound having six membered ring. Molecular studies provide evidence of presence of three alternative double bond providing exceptional stability to the molecule. Being basic unit of organic chemicals, benzene is widely used in pharmaceuticals, rubber, pesticides, dyes, lubricants, explosives and additives. Benzene derivatives are used in production of dyes, deodorants, pesticides, organic solvents, dispersants, plasticizers, and drugs. Keeping in view the extensive applications of benzene derivatives, we made use of QSPR analysis for predicting the multiple chemical and physical properties of organic compounds. By using SPSS software, correlation between newly introduced entropies and the Physico-chemical properties of benzene derivatives is examined. Our acquired consequences demonstrated that two physical properties of chemical structures, ie Boiling Point and enthalpy can be best predicted from the fifth ND entropy, and the other two properties namely Pie-electron Energy and molecular weight predicted from the neighborhood version of redefined first Zagreb entropy and neighborhood version of redefined second Zagreb entropy, respectively. [ABSTRACT FROM AUTHOR]

A chemical graph is a molecular complex that significantly explains the molecular structres. In this way atoms are identified by vertices and that are connected by lines known as edges of graph. Triangular benzenoids and starphene nanotubes are an inorganic compound with trigonal crystal system. We have computed the degree-based topological co-indices of triangular benzenoids and starphene nanotubes, that are successfully linked with its thermodynamic properties. [ABSTRACT FROM AUTHOR]

Let G be a simple, finite, undirected graph with n vertices. The main purpose of this paper introduces the concepts of the minimum covering Gutman Estrada index, the minimum covering Seidel Estrada index, the minimum covering distance Estrada index, the minimum covering Randić Estrada index, the minimum covering Harary Estrada index, and the minimum covering color Estrada index of a graph. First, we compute the new concepts for some of the graphs, such as cocktail party, star, crown, complete and complete bipartite. Moreover, we establish upper and lower bounds for the new concepts. [ABSTRACT FROM AUTHOR]

A variety of graphical invariants have been described and tested, offering lots of applications in the fields of nanochemistry, computational networks and in different scientific research areas. One commonly studied group of invariants is the topological index, which allows to research the chemical, biological, and physical properties of a chemical structure. Topological indexes are numerical quantities that can be used to describe the properties of the molecular graph. In this article, we draw from the analytically closed formulas of certain molecular structures of coronavirus such as Ribavirin, Sofosbuvir and Oseltamivir by calculating temperature based topological indices. [ABSTRACT FROM AUTHOR]