The main objective of the current research work was to synthesize mesoporous silica nanoparticles for controlled delivery of mometasone furoate for potential nasal delivery. The optimized sol–gel method was used for the synthesis of mesoporous silica nanoparticles. Synthesized nanoparticles were processed through Zeta sizer, SEM, TEM, FTIR, TGA, DSC, XRD, and BET analysis for structural characterization. The in vitro dissolution test was performed for the inclusion compound, while the Franz diffusion experiment was performed for permeability of formulation. For the determination of expression levels of anti-inflammatory cytokines IL-4 and IL-5, RNA extraction, reverse transcription, and polymerase chain reaction (RT-PCR) were performed. The MTT assay was also performed to determine cell viability. Synthesized and functionalized mesoporous silica nanoparticles showed controlled release of drugs. FT-IR spectroscopy confirmed the presence of the corresponding functional groups of drugs within mesoporous silica nanoparticles. Zeta sizer and thermal analysis confirmed the delivery system was in nano size and thermally stable. Moreover, a highly porous system was observed during SEM and TEM evaluation, and further it was confirmed by BET analysis. Greater cellular uptake with improved permeability characteristics was also observed. As compared to the crystalline drug, a significant improvement in the dissolution rate was observed. It was concluded that stable mesoporous silica nanoparticles with significant porosity were synthesized, efficiently delivering the loaded drug without any toxic effect.

Topological indices or coindices are one of the graph-theoretic tools which are widely used to study the different structural and chemical properties of the under study networks or graphs in the subject of computer science and chemistry, respectively. For these investigations, the operations of graphs always played an important role for the study of the complex networks under the various topological indices or coindices. In this paper, we determine bounds for the second Zagreb coindex of a well-known family of graphs called F -sum ( S -sum, R -sum, Q -sum, and T -sum) graphs in the form of Zagreb indices and coindices of their factor graphs, where these graphs are obtained by using four subdivision-related operations and Cartesian product of graphs. At the end, we illustrate the obtained results by providing the exact and bonded values of some specific F -sum graphs.

Topological indices or coindices are mathematical parameters which are widely used to investigate different properties of graphs. The operations on graphs play vital roles in the formation of new molecular graphs from the old ones. Let Γ be a graph we perform four operations which are S , R , Q ,  and  T and obtained subdivisions type graphs such that S Γ , R Γ , Q Γ , and T Γ , respectively. Let Γ 1 and Γ 2 be two simple graphs; then, F -sum graph is defined by performing the Cartesian product on F Γ 1 and Γ 2 ; mathematically, it is denoted by Γ 1 + F Γ 2 , where F ∈ S , R , Q , T . In this article, we have calculated sum-connectivity coindex for F -sum graphs. At the end, we have illustrated the results for particular F -sum graphs with the help of a table consisting of numerical values.

Let G = V E , E G be a connected graph with vertex set V G and edge set E G . For a graph G, the graphs S(G), R(G), Q(G), and T(G) are obtained by applying the four subdivisions related operations S, R, Q, and T, respectively. Further, for two connected graphs G 1 and G 2 , G 1 + F G 2 are F -sum graphs which are constructed with the help of Cartesian product of F G 1 and G 2 , where F ∈ S , R , Q , T . In this paper, we compute the lower and upper bounds for the first Zagreb coindex of these F -sum (S-sum, R-sum, Q-sum, and T-sum) graphs in the form of the first Zagreb indices and coincides of their basic graphs. At the end, we use linear regression modeling to find the best correlation among the obtained results for the thirteen physicochemical properties of the molecular structures such as boiling point, density, heat capacity at constant pressure, entropy, heat capacity at constant time, enthalpy of vaporization, acentric factor, standard enthalpy of vaporization, enthalpy of formation, octanol-water partition coefficient, standard enthalpy of formation, total surface area, and molar volume.

Topological indices are graph-theoretic parameters which are widely used in the subject of chemistry and computer science to predict the various chemical and structural properties of the graphs respectively. Let G be a graph; then, by performing subdivision-related operations S , Q , R , and T on G , the four new graphs S G (subdivision graph), Q G (edge-semitotal), R G (vertex-semitotal), and T G (total graph) are obtained, respectively. Furthermore, for two simple connected graphs G and H , we define F -sum graphs (denoted by G + F H ) which are obtained by Cartesian product of F G and H , where F ∈ S , R , Q , T . In this study, we determine first general Zagreb co-index of graphs under operations in the form of Zagreb indices and co-indices of their basic graphs.