On Entropy Measures and Eccentricity-Based Descriptors of Polyamidoamine (PAMAM) Dendrimers

Topological indices (TIs) assign a numeric value to a graph or a molecular structure. Due to their ability to predict the physiochemical properties of a molecular graph, several TIs have been introduced and studied, mainly based on degree and distance. For a vertex $v$ , the maximum distance of $v$ from any other vertex in a graph $G$ is called the eccentricity of $v$ , which is denoted by $\sigma \left(v\right)$ , in $G$ . The eccentricity of vertices gained special attention among the distance-based or degree-distance based TIs. An important TI in the class of eccentricity-dependent TIs is an eccentricity-entropy index. Furthermore, other eccentricity-dependent TIs such as eccentric-connectivity index, total-eccentricity index, and the first Zagreb index have also been extensively studied. On the other hand, dendrimers came out as unique polymeric macromolecules because of extensively branched three-dimensional architectural characteristics. This structure design prepares for various unique properties of dendrimers, including monodispersity, multivalency, uniform size, globular shape, water solubility with hydrophobic internal cavities, and a high degree of branching. These properties make them attractive candidates for different applications. PAMAM (polyamidoamine) dendrimers are promising polymers that can be successfully used in various biomedical applications. The PAMAM dendrimers having different structures such as a primary amine as the end group or porphyrin core have been studied through graph-theoretic parameters. This paper studies these two types of PAMAM dendrimers through eccentricity-dependent parameters. In particular, we establish formulae of eccentricity entropy for two types of PAMAM dendrimers. Moreover, we also derive analytic formulae of some other significant TIs from the class of eccentricity-dependent TIs. Furthermore, we apply graphical tools to demonstrate the trends of the values in the obtained results.

On Ve-Degree-Based Irregularity Properties of the Crystallographic Structure of Molecules

Irregularity indices are usually used for quantitative characterization of the topological structure of nonregular graphs. In numerous applications and problems in material engineering and chemistry, it is useful to be aware that how irregular a molecular structure is? In this paper, we are interested in formulating closed forms of irregularity measures of some of the crystallographic structures of ${\text{Cu}}_2\text{O}\left[p,q,r\right]$ and crystallographic structure of titanium difluoride of $T_i{F}_2\left[p,q,r\right]$ . These theoretical conclusions provide practical guiding significance for pharmaceutical engineering and complex network and quantify the degree of folding of long organic molecules.