Topological indices are numbers calculated from conventional formulas obtained after applying mathematical operations on the graphs using various methods. The need for using tools like topological indices arised from the fact that physical and chemical attributes are given as numbers which have further given them a measure that allowed for comparisons and correlations. These descriptors are crucial tools in QSAR/QSPR research because they serve as molecular descriptors. The Shannon’s entropy concept-inspired graph entropies with topological indices to act as the units of information for calculating the structural information of chemical graphs and complicated networks. Discrete mathematics, biology, and chemistry are just a few of the fields where the graph entropy measures are crucial
In this study, various degree based molecular descriptors of two variations of thiophene dendrimers and their corresponding entropies have been calculated.
We examine a simple, finite, connected graph G with a set of vertex values of V and a set of edge values of E. The total number of edges that are incident on a vertex
where
The entropy measure is denoted by
|
|
First Zagreb |
|
Second Zagreb |
|
Harmonic |
|
Hyper Zagreb |
|
Forgotten |
|
Randic |
|
Reciprocal Randic |
|
Sum-connectivity index |
|
Geometric arithmetic |
|
Atom bond connectivity |
|
Irregularity measure |
|
Sigma |
|
In this section, we compute the degree based indices in
Proof. The graph
Edge partition |
Number of edges |
|
|
|
|
|
|
|
|
|
|
Let ;
Proof.
Let ;
Using Theorem 1 and by applying equation 1, we obtain the following;
Let
Similarly, For
For
For
For
For
For
For
For
For
For
For
On substituting the values and simplifying the equations, we get the result.
Proof. The graph
can be
which are
and the corresponding number of edges in each partition is given in
Edge partition |
Number of edges |
|
|
|
|
|
|
Proof. Using Equation 1, Theorem 3 and
A graphical comparison of the degree based descriptors and entropy measures of thiophene dendrimers have been obtained in this section.
Molecular descriptors, which are inextricably linked to the idea of molecular structure, play a critical role in scientific study, serving as the theoretical foundation of a complex network of information. The distinctive characteristics of a compound's physical, chemical and biological properties are determined by its structure. Topological indices are two-dimensional descriptors that are obtained from the structural representation of molecules. These descriptors are most usefully applied in QSAR and QSPR analysis. The molecular mechanism of a chemical may be inferred from its structure using a mathematical function called a QSAR model. The main phases in QSAR analysis are model generation, model validation, and its interpretation. Another crucial characteristic of descriptors is their Shannon entropy distribution, since those with larger entropies are predicted to provide more insightful prediction models. A method for evaluating the distribution and information content stored in descriptors is the Shannon entropy. The associated importance of this method is found in the determination of the physical implications of the graph-theoretical descriptors for Quantitative Structure-Property and Structure-Activity relationships in branching tree-like polymers. The Shannon's entropy is also frequently used to evaluate the variety of chemical resources and to analyze the information content of molecular descriptors within data sets of molecules. These descriptors and accompanied entropy measures serve as a gate way for further experimental research work. Future scope of research can be extended in formulating various other attributes including distance based indices, neighborhood degree indices, temperature indices, M-polynomial approach for certain indices, reverse degree based indices and a great variety of descriptors, for the family of thiophene dendrimers.
We have used Shannon's approach in finding the entropy measures of these thiophene dendrimers from their corresponding degree based molecular indices values. The graphical comparison between various entropy measures and also the indices values of these structures have also been portayed. Since the degree-based entropy has several possibilities in several fields of study, including computer science, chemistry, pharmaceuticals and biological therapies, a significant amount of research has to be carried out for enhancing the topological attributes of various chemical graphs. Accordingly, scientists can benefit from these numerical and graphical interpretation for various analysis of the structure. For a better understanding of the topic, several entropy measures and molecular indices have also been analyzed graphically. The neighborhood degree based indices using m polynomials and corresponding entropies can also be evaluated as a part of further research in this topic. Future study can extend this concept to different chemical structures.