On analysis of phthalocyanine network through statistical method

Any chemical network’s modeling and design heavily rely on graph theory. Chemical applications of graph theory determine a wide range of parameters, including thermodynamic properties, physico-chemical properties, chemical activity, and biological activity. Topological indices, which are specific graph invariants, can be used to characterize these features¹. A subfield of mathematical chemistry known as chemical graph theory uses graph theory techniques to model chemical phenomena quantitatively. It also has to do with the nontrivial uses of graph theory in molecular problem-solving. Let G be a graph, with the vertex set V(G) and edge set E(G). A vertex’s degree, or the number of connections to it, is represented by \(\Lambda _{\mu }\). Graphs can be further categorized as directed graphs (digraphs), weighted graphs, or bipartite graphs based on these features. All these graph types are useful for describing and analyzing complex systems in many fields, including computer science, social networks, biology, and transportation². Molecular graph theory and computational techniques enable the investigation of many molecular features, such as electronegativity, topological indices, and molecular weight. This facilitates the creation of novel compounds with the appropriate qualities and the comprehension of molecular-level chemical reactions. Essentially, chemical and molecular graphs are the foundation of modern chemistry and its practical applications.

Chemical graph theory makes use of topological indices, which are numerical values or descriptors that describe the structural properties of molecules. These indices are derived only from the connections between atoms inside a molecular graph, disregarding the spatial arrangement of atoms³. They are crucial to many branches of chemistry and fulfill a number of important functions. Topological indices are frequently utilized in QSPR research to connect a molecule’s structural characteristics with physicochemical characteristics. This helps in predicting the properties of a molecule, including solubilities, boiling points, and biological activity. Pharmaceutical research benefits from the development of new drug molecules thanks to topological indices. In Gutman,⁴, the degree-based topological indices were calculated. The multiplicative Zagreb indices and Zagreb indices of Eulerian graphs were calculated by Liu et al.⁵. The revised Zagreb indices of some cycle-related graphs and linear [n]-Havare, Ö, discussed phenylenes. Ç⁶. The generalized operations of graphs through Zagreb indices that are linked to subdivision were examined by Liu et al.⁷. Using a polynomial approach, Lal et al.⁸ calculated the topological indices of lead sulfide. Kirana et al.⁹ used QSPR and curvilinear regression to analyze the Quinolone antibiotics for different degree-based topological metrics. Sohan Lal examined the graph entropies and topological indices for carbon nanotube Y-junctions, et al.⁸. Topological indices, especially indices, are crucial for determining relationships between molecular structures and their therapeutic effectiveness, according to Siddiqui et al.^10,11. By employing underlying molecular networks to analyze MOFs, Nadeem et al.¹² are able to extract and investigate the topological characteristics of these intricate structures. The work emphasizes how useful topological metrics are for comprehending MOF connection, stability, and function. According to topological descriptors, Ahmed et al.¹³ concentrate on using supervised machine learning methods to forecast the physicochemical characteristics of anti-HIV medications. Ahmad^14,15,16 investigate the degree based topological indices of benzene ring embedded in P-type-surface in 2D network. A comparative investigation on valency-based topological descriptors for the Hexagon Star is carried out by Koam et al.¹⁷.

Hayat et al.¹⁸ analyzed the topological descriptor for certain graphs. Researchers can make educated decisions about changing molecular structures to enhance therapeutic efficacy and lessen negative effects by examining the topological characteristics of existing pharmaceuticals and their targets¹⁹. Topological indices can be used in materials research to better understand the structure-property interactions in a variety of materials, including polymers, catalysts, and nanomaterials. This helps create materials with the desired qualities. Gutman, and Estrada²⁰ computed the molecular descriptor for line graphs. Khalaf et al.²¹ discussed the degree-based indices for bridge graphs. Furtula, B.,²² analyzed the structure using the topological indices. Darafsheh, M. R.²³ computed the topological descriptor for some graphs. Different Zagreb-type topological indices²⁴ are defined in Table 1.

Dehmer et al.²⁵ discussed the history of entropy measure. Manzoor et al²⁶ computed the entropy measures of molecular graphs. Arockiaraj et al.²⁷ analyzed the entropy of zeolites BCT and DFT with bond-wise scaled comparison. Naeem et al.²⁸ discussed the regression model for entropy. Shao et al.²⁹ discussed the different descriptors for symmetrical nanotubes. Zuo et al.^30,31 computed the Shanon entropy of Benzenoid Systems. Zhao et al.³² discussed the entropy of silver iodide.

Rashevsky³³, Mowshowitz³⁴ and Chen et al.³⁵ introduced the concept of entropy in graphs.

The formula for entropy is shown in Eq. (1).

The edge weight of the edge in G is shown in this graph as \(\Lambda (\mu \nu )\). Hanif et al.³⁶ analyzed the entropy measure for vanadium III chloride. Xavier et al.^37,38 computed the degree-based entropy for Thiophene Dendrimers. Kazemi³⁹ computed the entropy measure for some molecular graphs using degree-based indices. Chu et al.⁴⁰ discussed the curve fitting between entropy and indices for graphite carbon nitride. Ma et al.^41,42 analyzed for graphitic carbon nitride via enthalpy and entropy measurements.

A remarkable family of materials that has received a lot of attention in the fields of materials science and nanotechnology is called two-dimensional transition metal phthalocyanine (2D-TMPC). These substances are made of phthalocyanine molecules, which have a big, planar, aromatic structure made up of four isoindole rings joined by nitrogen atoms⁴³. In the presence of nitrogen atoms in the phthalocyanine framework, transition metal atoms like iron (Fe), copper (Cu), or cobalt (Co) can coordinate to produce 2D-TMPCs^44,45. Two common methods for creating 2D-TMPCs are chemical vapour deposition (CVD), molecular beam epitaxy (MBE), or solution-based methods, in which the transition metal atoms are introduced during the growing process.

Figure 1 shows the 2D TM-Pc sheet in the \(1\times 1\) and \(3\times 3\) network. The unit cell is shown by a square on the left side. The blue colours of vertices show carbon, the black colours show hydrogen, the pink colours show nitrogen, and the green colours show metal.

2D TM-Pc network⁴³.

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Using figure, we have obtained the edge partition of our graph as presented in the table, which is one of the major steps in our research. Further, in this section, we extend our research by computing the Zagreb-type indices for our graph. In fact, such types of indices have been employed for understanding several topological qualities and structural traits of any graph. Not only their computation but we also present their values in graphical and numerical fashion. By doing so, this serves as a holistic approach wherein insights to the underlying patterns and linkages in the graph are obtained, thus enhancing further understanding of the behavior of the graph and its relevance to our research.

• The Bi-Zagreb index is found using Table 2 in Table 1.

• We have the Tri-Zagreb index using Table 2 in Table 1.

• Table 2 in Table 1 gives us the following geometric-bi-Zagreb index.

• The geometric-tri-Zagreb index is obtained using Table 2 in Table 1.

• We obtain the geometric-harmonic index by using Table 2 in Table 1.

• We have the harmonic-bi-Zagreb index using Table 2 in Table 1.

• We have the harmonic-tri-zagreb index using Table 2 in Table 1.

• The harmonic geometric index can be obtained by using Table 2 in Table 1.

• We have the bi-Zagreb-geometric index using Table 2 in Table 1.

• We have the bi-Zagreb-harmonic index using Table 2 in Table 1.

• For Table 1, we have the tri-Zagreb-harmonic index using Table 2.

• For Table 1, we have the tri-Zagreb-geometric index using Table 2.

Table 3 illustrates that when m and n are elevated, the indices expand rapidly based on a comparison of their numerical values. This result demonstrates how susceptible the indices are to variations in these parameters.

Comparison between (a) GH(G), BM(G), TM(G)     (b) (a) HTM(G), HBM(G), BMG(G).

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Figure 2a shows an interesting tendency in that the Tri-Zagreb index increases noticeably more quickly than the BM and GH indices. This finding emphasizes the Tri-Zagreb index’s clear sensitivity to changes in the underlying data, which may indicate certain structural traits. In Fig. 2b, we see a similar pattern where BMG shows a marked and quick increase in comparison to HTM and HBM. This raises the possibility that the BMG index may be useful in identifying particular structural patterns or features in the data being studied because it suggests that the index is more sensitive to the factors affecting its calculation. These results underline how crucial it is to pick indices that are compatible with the distinct traits and characteristics of the data being examined.

Comparison between (a) GTM(G), HM(G), GHB(G)     (b) TMG(G), BMH(G), TMH(G).

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Figure 3a makes it evident that the GBM index rises at a substantially faster rate than the HG and GTM indices. This unique behaviour highlights the intrinsic differences in their growth patterns and suggests that the GBM index may be more sensitive or reactive. As with the previous point, Fig. 3b clearly shows that TMH grows at a faster pace than TMG and BMH. These variations in growth trajectories demonstrate how important it is to understand the special characteristics and behaviour of these metrics. Comparing them can provide insights into their underlying mechanisms and implications.

We computed entropy for several indices.The Bi-Zagreb entropy is obtained by using Table 2 and Eq. (2) in Eq. (1).

Using Table 2 and Eq. (3) in Eq. (1) we obtain the Tri-Zagreb entropy.

Using Table 2 and Eq. (4) in Eq. (1) we obtain the geometric-bi-Zagreb entropy.

Using Table 2 and Eq. (5) in Eq. (1) we obtain the geometric-tri-Zagreb entropy.

Using Table 2 and Eq. (6) in Eq. (1) we obtain the Geometric- Harmonic entropy.

Using Table 2 and Eq. (7) in Eq. (1) we obtain the harmonic-bi-Zagreb entropy.

Using Table 2 and Eq. (8) in Eq. (1) we obtain the harmonic-tri-Zagreb entropy.

Using Table 2 and Eq. (9) in Eq. (1) we obtain the Harmonic- Geometric entropy.

Using Table 2 and Eq. (10) in Eq. (1) we obtain the bi-Zagreb-geometric entropy.

Using Table 2 and Eq. (11) in Eq. (1) we obtain the bi-Zagreb-geometric entropy.

Using Table 2 and Eq. (12) in Eq. (1) we obtain the bi-Zagreb-harmonic entropy.

Using Table 2 and Eq. (13) in Eq. (1) we obtain the tri-Zagreb-harmonic entropy.

We notice that the following entropy values for varying the m and n parameters, as shown in Table 4, is a striking trend. From this table we readily observe that starting to go from 3 to 8 the entropy values increase rather rapidly. This rapid growth reflects that the greater the values of m and n the more unpredictable and disordered the system becomes. The entropy of the system, which equally grows with such parameters, indicates complicacy and, therefore, requires further understanding of how changes in these parameters do affect the behavior of the system studied. This will equally provide the necessary insight for making decisions, which are accurate and forecasts within the range of parameters constituting a system under study.

Comparison between (a) \(ENT_{GH}(G), ENT_{BM}(G), ENT_{TM}(G)\)     (b) \(ENT_{GTM}(G), ENT_{HM}(G), ENT_{GHB}(G)\).

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That ENTBM is increasing substantially faster than that of ENTTM and ENTGH can be seen from Fig. 4a, if one look carefully. This implies that ENTBM is more sensitive to fluctuations and uncertainties within the system. Figure 4b plots ENTGBM increasing entropy at a rate much faster than ENTGTM and ENTHG. The differences in the rates of increase of entropy underscore the unique behavior of these variables and also point to the special roles these variables play in the system.

Comparison between (a) \(ENT_{HTM}(G), ENT_{HBM}(G), ENT_{BMG}(G)\)     (b) \(ENT_{TMG(G)}, ENT_{BMH}(G), ENT_{TMH}(G)\).

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In fact, as can be seen from Fig. 5a, \(ENT_{BMG}\) is increasing much faster compared to \(ENT_{HTM}(G)\) and \(ENT_{HBM}(G)\). While its growth rate is different from the sensitivity of the system to the \(ENT_{BMG}(G)\), this creates a belief that it is more sensitive to changes and uncertainties. Similarly, Fig. 5b also confirms the previously made assertion that \(ENT_{TMG(G)}\) is rising much faster compared to both \(ENT_{BMH}(G)\) and \(ENT_{TMH}(G)\) in terms of entropy. These divergences in the growth rate of entropy illustrate peculiar features of these variables and give clues about specific works each of them performs in the system. Prudent decisions on management or analysis, and gaining insight into the dynamics of a system, rest on advanced knowledge of these different responses.

A statistical measure of the linear relationship between two variables is the Pearson correlation. A perfect negative correlation is represented by a number between \(-1\) and 1, a perfect positive correlation by a number between 1 and 0, and no correlation at all by a number between 0 and 1. To get the Pearson correlation, divide the covariance of the two variables by the sum of their standard deviations. It is simpler to compare the strength of correlations between various variables when using Pearson correlation, as it is a normalized measure of covariance. Because it is fairly easy to calculate and comprehend, Pearson correlation is a widely used statistical approach.

Pearson correlation between indices and entropy.

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The Pearson correlation heat map as shown in Fig. 6 suggests that between the indices and entropy measures, there exists a consistently strong positive linear relationship, as most of the values in this area fall around 0.86 to 0.87. That could imply that, very possibly, a similar mathematical or structural pattern underlies the indices and the entropy measures. Namely, HTM(G) and HG(G) are slightly higher than 0.87 with the respective entropy measures, \(ENT_HTM\) and \(ENT_HG\), respectively, which reflects stronger alignment in their relationships. There is a lack of negative correlations in the sense of indicating that for those tending to increase, so do the entropy measures; this further reflects directional behavior in an aligned way. However, the narrow spread of associations indicates a lack of distinctiveness among the latter, which is arguably indicative of the presence of high multicollinearity or the commonality of some underlying properties among the series. These would indicate that the indices and entropy measures may be derived from common backgrounds or are similarly influenced by dominant common factors. More analyses, such as regression or even the dimension reduction technique PCA, can be conducted for the selection of the most predictive indices and hence reducing redundancy.

In this paper, we conducted a thorough investigation of different indices and used them to compute entropy. As part of our investigation, we also looked into the Pearson correlation coefficients to see how they related indices to entropy. The results of our study show that there is a strong and substantial relationship between the computed indices and entropy, with a correlation coefficient of roughly 0.87. The interdependence between these variables is shown by the significant positive correlation, which implies that changes in the indices are closely related to changes in entropy. The usefulness of these indices as indicators or predictors of entropy-related events is highlighted by this observation, which sheds light on the complex dynamics of the system under study.