Mathematical modeling with the help of numerical coding of graphs has been used in the different fields of science, especially in chemistry for the studies of the molecular structures. It also plays a vital role in the study of the quantitative structure activities relationship (QSAR) and quantitative structure properties relationship (QSPR) models. Todeshine et al. (2010) and Eliasi et al. (2012) defined two different versions of the 1st multiplicative Zagreb index as ∏Γ=∏p∈VΓdΓp2 and ∏1Γ=∏pq∈EΓdΓp+dΓq, respectively. In the same paper of Todeshine, they also defined the 2nd multiplicative Zagreb index as ∏2Γ=∏pq∈EΓdΓp×dΓq. Recently, Liu et al. [IEEE Access; 7(2019); 105479–-105488] defined the generalized subdivision-related operations of graphs and obtained the generalized F-sum graphs using these operations. They also computed the first and second Zagreb indices of the newly defined generalized F-sum graphs. In this paper, we extend this study and compute the upper bonds of the first multiplicative Zagreb and second multiplicative Zagreb indices of the generalized F-sum graphs. At the end, some particular results as applications of the obtained results for alkane are also included.

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Metal-organic networks (MONs) is a family of chemical compounds consisting of clusters or metal ions and organic ligands. These are studied as one, two or three dimensional structures of porous materials and subclasses of coordination polymers. MONs are mostly used in catalysis for the separation & purification of gases and as conducting solids or super-capacitors. In some situations, these networks are found to be stable in the process of removal or solvent of the guest molecules and could be restored with some other chemical compounds. The physical stability and mechanical properties of these networks have become a topic of great interest due to the aforesaid characteristics. Topological indices (TIs) are numeric quantities that are used to forecast the natural relationships among the physico-chemical characteristics of the chemical compounds in their fundamental network. During the studies of the MONs, TIs show an essential role in the theoretical & environmental chemistry and pharmacology. In this paper, we compute various latest developed degree-based TIs for two different metal-organic networks with increasing number of layers consisting on both metal and organic ligands vertices as well. A comparison among the computed different versions of the TIs with the help of the numerical values and their graphs is also included. INDEX TERMS Topological indices, chemical compounds, metals-organic networks.

COVID-19 spread rapidly and jeopardised the physicians with not only treatment but also with limited resources and new working style. This study aimed to investigate the experience of physicians in Pakistan as frontline workers for the treatment of COVID-19 patients. The study employed a qualitative design and used Haas's adaptation of Colaizzi method for analysis. The respondents for the interview were selected based on purposive sampling and only those physicians were contacted who were treating COVID-19 patients in Lahore, Pakistan. The results of the study revealed many challenges faced by Pakistani physicians which included physical and psychological stress due to the treatment of COVID-19 patients. Moreover, infrastructural flaws have added in the vulnerabilities of the physicians. In developing countries, the fight is much harder for physicians as evidenced by their first-hand experience. The policymakers in developing countries especially in Pakistan can direct policies facilitating physicians to reduce their physical and psychological stress as well as increasing resources for the treatment of COVID-19 patients.

Metal-organic frameworks (MOFs) are porous materials formed by strong bonds between metal ions and organic ligands to represent very high surface area, large pore volume, excellent chemical stability and unique morphology. Work on synthesis, structures and characteristics of many MOFs shows the importance of these frameworks with versatile applications, such as energy storage devices of excellent electrode materials, gas storage, heterogeneous catalysis, environmental hazard, assessment of chemicals and sensing of different gases. A topological property or index is a numerical invariant that predicts the physicochemical properties of the chemical compounds of the underlying molecular graph or framework. Wiener (1947) created the practice of the topological indices (TI’s) in organic molecules with the reference of boiling point of paraffin. In this paper, we study the two different metal-organic frameworks with respect to the number of increasing layers with metal and organic ligands as well. We also compute the generalized Zagreb index and generalized Zagreb connection index of these frameworks. Moreover, the various indices and connection indices are obtained by using the aforesaid generalized versions. At the end, a comparison is also included between the indices and connection indices with the help of numerical values and their 3D plots.

The first general Zagreb (FGZ) index (also known as the general zeroth-order Randić index) of a graph G can be defined as M γ G = ∑ u v ∈ E G d G γ − 1 u + d G γ − 1 v , where γ is a real number. As M γ G is equal to the order and size of G when γ = 0 and γ = 1 , respectively, γ is usually assumed to be different from 0 to 1. In this paper, for every integer γ ≥ 2 , the FGZ index M γ is computed for the generalized F-sums graphs which are obtained by applying the different operations of subdivision and Cartesian product. The obtained results can be considered as the generalizations of the results appeared in (IEEE Access; 7 (2019) 47494–47502) and (IEEE Access 7 (2019) 105479–105488).

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The combination of mathematical sciences, physical chemistry, and information sciences leads to a modern field known as cheminformatics. It shows a mathematical relationship between a property and structural attributes of different types of chemicals called quantitative-structures’ activity and qualitative-structures’ property relationships that are utilized to forecast the chemical sciences and biological properties, in the field of engineering and technology. Graph theory has originated a significant usage in the field of physical chemistry and mathematics that is famous as chemical graph theory. The computing of topological indices (TIs) is a new topic of chemical graphs that associates many physiochemical characteristics of the fundamental organic compounds. In this paper, we used the M-polynomial-based TIs such as 1st Zagreb, 2nd Zagreb, modified 2nd Zagreb, symmetric division deg, general Randi c ´ , inverse sum, harmonic, and augmented indices to study the chemical structures of pent-heptagonal nanosheets of V C 5 C 7 and H C 5 C 7 . An estimation among the computed TIs with the help of numerical results is also presented.