There is a long-standing connection between chemistry and graph theory, which ultimately led to the development of chemical graph theory. The strong relationship between these two fields laid the foundation for this interdisciplinary specialty. Isaac Newton died in 1727, but his theoretical contributions to physics and mathematics helped lay the foundations that influenced later developments in chemistry. In the 18th century, scientists began studying how atoms interact, and later in the century the first chemical graphs were created to show the relationships between atoms and molecules.1, 2.
Chemical graph theory has found wide applications in various fields such as materials science, drug discovery, QSPR/QSAR research, and chemical synthesis.3. In graph theory, a graph is \(G=(V,E)\) Consists of a series of points (vertices) V and a series of lines (edges) E. A chemical graph shows how compounds are structured and organized. Bonds between atoms are lines, and atoms are nodes. Graph theory can model and characterize complex systems and interactions, providing practical solutions for structural analysis, optimization, as well as integrating multiple fields of technology and science.4,5.
Topological measurements (also called structural invariants of molecular graphs) quantify the connectivity of molecular networks. These indices have recently gained popularity due to their use in quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR). In this case, order-based topological indicators have been extensively studied and applied to predict the physicochemical properties of molecular structures.6, 7, 8.
The literature shows numerous reports describing topological indexes, and their usefulness can be found in various research fields such as computer science, physics, biology, and chemistry, in addition to drug development. The oldest and most widely studied topological metric is the Wiener metric, introduced by Wiener in 1947.9. This refers to the construction of the minimum sum of the lengths of the shortest paths between each vertex of the molecular graph. It has also been used to predict the boiling point of paraffin, one of its most notable applications.
Several recent studies have investigated the creation and use of new degree-based topological indices. Masmari et al.10 defined a mathematical model for the irregularity index of adriamycin and performed a statistical analysis of emetics. Rasheed et al.11 A unique index was used to study the properties of octane isomers. Rai et al.12 evaluated an M polynomial-based topological index between networks built with subdivision and polyhex structures. Abirami et al.13 We calculated the order-based index of the complex molecular structure of ruthenium bipyridine. Huang et al.14 investigated the correlation between entropy and topological index of silicon dioxide using a regression model. in15 We discuss computational and comparative aspects of two carbon nanosheets with respect to several new topological metrics. This study focuses on the computation and analysis of order-based topological indices.
The Zagreb index, established by Gutman and Trinajstić in 1972, is an important topological descriptor in chemical graph theory.16. Quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) frameworks have widely used these metrics for modeling molecular properties such as stability, boiling point, and biological activity, among others. The Zagreb index of triangular boron nanotubes and the line graph of triangular boron nanotubes were calculated as follows.17,18.
Let’s consider a graph G and the top \(s \in V(G)\). degree sindicated by \(\degree_s\)is defined as the number of edges that match the next edge. s. The Zagreb index is defined as:
$$\begin{aligned} M_1(G)&=\sum _{st\in E(G)}(\deg _s+ \deg _t),\\ M_2(G)&=\sum _{st\in E(G)}(\deg _s\times \deg _t). \end{Align}$$
Another common degree-based topology descriptor is the Randić index, also known as the connectivity index. Introduced by Mr. Landich19 In 1975, it was given by
$$\begin{aligned} R_\alpha (G) = \sum _{st\in E(G)}\left( \deg _s \times \deg _t\right) ^\alpha , \end{aligned}$$
where \(\alpha\) is a real number. research such as20,21 They calculated the Landich index for various chemical structures and highlighted its application in predicting the stability, boiling point, and other physical properties of molecules.
In 1988, Estrada et al.twenty two introduced the atomic bond connectivity index, which is a topological index in chemical graph theory. The atomic bond connectivity index is defined as:
$$\begin{aligned} ABC(G)=\sum _{st\in E(G)}\sqrt{\frac{\deg _s+\deg _t-2}{\deg _s \times \deg _t}}, \end{aligned}$$
It was used to model the stability of alkanes. This shows a strong correlation with catalyst durability and chemical reactivity. Das et al.twenty four Proven results for the maximum general ABC index of graphs with a given maximum degree.
Vukicevic and Furturatwenty three In 2009, we proposed the Geometric Arithmetic (GA) index.
$$\begin{aligned} GA(G)=\sum _{st\in E(G)}{\frac{2\sqrt{\deg _s \times \deg _t}}{\deg _s+\deg _t}}. \end{Align}$$
Zhou and Trinajstić introduced the total connectivity index in 2009twenty five. it is defined as
$$\begin{aligned} SCI(G)=\sum _{st\in E(G)}\frac{1}{\sqrt{\deg _s + \deg _t}}. \end{Align}$$
The total connectivity index measures the degree of interconnection between atoms in a chemical network. Helps analyze molecular behavior and structural complexity in biological and chemical systems.
Entropy has emerged as an important analytical tool across a variety of scientific fields, including mathematics, graph theory, and computational modeling. Entropy was first applied to environmental research, but Haynes et al.26 We highlighted its increasing use in fields such as engineering and social sciences and demonstrated its versatility in modeling complex systems. Zhou et al.27 Iron(II) chloride was analyzed using statistical models and graph entropy metrics. Ovais et al.28 We calculated entropy measurements for the dendrimers using a degree-based index. Kunimine et al.29 We investigated the entropy metrics and topological indices of the berylonitrene network through a logarithmic regression model. Irfan et al.30 We investigated entropy measurements in polymer graphs using a logarithmic regression model. Mondal et al.31 We presented the role of entropy in structural property modeling of molecules. recent articles 32,33and34 We describe the application of Sombor topological index and entropy measurements to QSPR modeling of anticancer drugs (Python-based methodology), information-theoretic entropy, and topological descriptor analysis of tin oxide. \((SnO_{2})\) They are used for drug structure and property prediction using machine learning, molecular graph and entropy-based QSPR analysis, respectively.
Entropy has been widely studied in network science and graph theory using various topological metrics that capture structural properties of graphs. The integration of entropy into topological analysis can be traced back to early information theory, where Shannon first introduced the concept of entropy.35
$$\begin{aligned} ENT=-\sum _{i=1}^{r}U_i\frac{f(\deg _s\deg _t)}{TI}\log \frac{f(\deg _s\deg _t)}{TI} =\log (TI)-\frac{1}{TI}\sum _{i=1}^{r}U_i{f(\deg _s\deg _t)}\log f(\deg _s\deg _t). \end{Align}$$
In this context, \(TI = \sum _{i=1}^{r} U_i f(\deg _s \deg _t)\) Represents a topology index. Here, \(U_i\) is the frequency, r is the number of edges, \(f(\degree_s \degree_t)\) Weight of edge connecting vertices in degrees \(\degree_second\) and \(\degree_t\).
