The analysis of networks and graphs through topological descriptors carries out a useful role to derive their underlying topologies. This process has been widely used in biomedicine, cheminformatics, and bioinformatics, where assessments based on graph invariants have been made available for effectively communicating with the various challenging schemes. In the studies of quantitative structure-activity relationships (QSARs) and quantitative structure-property relationships (QSPRs), graph invariants are used to approximate the biological activities and properties of chemical compounds. In this paper, we give the results related to the eccentric-connectivity index, connective eccentricity index, total-eccentricity index, average eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmetic index, eccentric atom-bond connectivity index, eccentric adjacency index, modified eccentric-connectivity index, eccentric distance sum, Wiener index, Harary index, hyper-Wiener index and degree distance index of a new graph operation named as “subdivision vertex-edge join” of three graphs.

It is revealed from the previous studies that there is a strong relation between the chemical characteristic of a chemical compound and its molecular structure. Topological indices defined on the molecular structure of biomolecules can help to gain a better understanding of their physical features and biological activities. Eccentricity connectivity indices are distance-based molecular structure descriptors that have been used for the mathematical modeling of biological activities of diverse nature. As the porphyrin has photofunctional properties, such as a large absorption cross-section, fluorescence emission, and photosensitizing properties, due to these properties, porphyrin dendrimers can be used as photofunctional nanodevices. In this paper, we compute the exact formulae of different versions of eccentric connectivity index and their corresponding polynomials for a class of porphyrin-cored dendrimers. The results obtained can be used in computer-aided molecular design methods applied to pharmaceutical engineering.

Several graph invariants have been defined and studied, which present applications in nanochemistry, computer networks, and other areas of science. One vastly studied class of the graph invariants is the class of the topological indices, which helps in the studies of chemical, biological, and physical properties of a chemical structure. One recently introduced graph invariant is the face index, which can assist in predicting the energy and the boiling points of the certain chemical structures. In this paper, we drive the analytical closed formulas of face index of silicon carbides Si2C3−Ia,b, Si2C3−IIa,b, Si2C3−IIIa,b, and SiC3−IIIa,b.

In the study of the quantitative structure–activity relationship and quantitative structure-property relationships, the eccentric-connectivity index has a very important place among the other topological descriptors due to its high degree of predictability for pharmaceutical properties. In this paper, we compute the exact formulas of the eccentric-connectivity index and its corresponding polynomial, the total eccentric-connectivity index and its corresponding polynomial, the first Zagreb eccentricity index, the augmented eccentric-connectivity index, and the modified eccentric-connectivity index and its corresponding polynomial for a class of phosphorus containing dendrimers.

A graph is said to be a regular graph if all its vertices have the same degree, otherwise, it is irregular. Irregularity indices are usually used for quantitative characterization of the topological structure of non-regular graphs. In numerous applications and problems in material engineering and chemistry, it is useful to be aware that how irregular a molecular structure is? Furthermore, evaluations of the irregularity of underline molecular graphs could be valuable for QSAR/QSPR studies, and for the expressive determines of chemical and physical properties, such as enthalpy of vaporization, toxicity, resistance, Entropy, melting and boiling points. In this paper, we think over the following four irregularity measures: the irregularity index by Albertson, σ irregularity index, the total irregularity index and the variance of vertex degrees. By way of graph structural estimation and derivations, we determine these irregularity measures of the molecular graphs of different classes of dendrimers.