Introduction

Chemical graph theory examines and comprehends chemical compounds using graph theory principles. In this context, atoms are called vertices and chemical bonds between atoms are called edges. The arrangement and connections between the atoms in a molecule can be shown using chemical graphs. By applying graph theory concepts like as degrees, order, and size, investigators may investigate the physical characteristics of chemical compounds and get more understanding into their reactivity, stability, and other chemical properties. A topological descriptor is a numerical value derived from graph theory that represents the structural features of a chemical compound. When a topological descriptor exhibits correlation with a molecular attribute, it can be expressed as either a topological index or a molecular index. The thermodynamic properties (such as boiling points, heat of combustion, enthalpy of formation, etc.) and a number of other properties showed good association with the structure. As a result, a topological index changes a chemical structure into a specific number that is useful for QSPR/QSAR research. Havare et al.1 illustrated the characteristics of novel drugs employed for cancer treatment using QSPR modeling and topological indices. Zhong et al.1 explored the quantitative structure–property relationships (QSPR) valency-based topological indices with COVID-19 pharmaceuticals. Generalized multiplicative first Zagreb index was computed by Hayat et al.2 and applied to graph QSPR modeling. For nanotubes, Zhang et al.3 Calculated the topological indices. Regression modeling was employed by Zaman et al.4 to analyze QSPR for Drugs Used in Blood Cancer Treatment. Several investigations have been initiated to investigate the efficacy of well-known indices, such as the Wiener, Zagreb, and Randic indices, in predicting the properties of molecules5,6,7,8,9,10,11,12,13,14,15,16,17,18,19. It has been demonstrated that these indices are highly beneficial when used in drug design, quantitative structure–property relationships (QSPR), and recognizing the intricate links between molecular structures and functioning20,21,22,23,24,25,26,27,28,29,30,31,32,33,34. Table 1 displays these topological indices.

Table 1 Degree based topological descriptors.
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The concept of entropy was first introduced by Shannon43. It determines how unpredictable the information content of a system is. It has been successfully utilized to investigate chemical networks and graphs. The graph entropy was introduced by Rashevsky44 using the classification of vertex orbits and was introduced in 1955. These days, graph entropy is used in numerous scientific disciplines, including chemistry and biology45. Manzoor et al.46 discovered the molecular graphs’ entropy metrics. The extremity of degree-based graph entropies was covered by Cao et al.47. Dehmer et al. investigated the background of graph entropy metrics48. Galavant et al.49 discussed about entropy based on the first degree. Liu J-B et al.50 explored Octahedron networks and determined topological indices based on degree. Asad et al.28 studied the structural complexity and irregularity of Kudriavite (CdBi2S4) using topological analysis. The predictive ability of entropy measures based on both multiplicative descriptor versions was examined by Paul D. et al.11. Ullah et al.51 investigated Network-Based Modeling of the Molecular Topology of Fuchsine Acid Dye with Respect to Certain Irregular Molecular Descriptors. Degree-based and reverse degree-based irregularity indices for the sodalite material network were modeled and characterized in three dimensions by Zaman et al.52. Ullah et al.53 explored the development of various bioconjugate networks and their structural modeling using irregularity topological indicators. Similarly, numerous other studies investigated the structural characteristics and structure–property relationships in various material systems by using different kinds of topological indices54,55,56,57,58,59,60,61,62,63,64,65,66,67.

Let \(\Re\) be an edge-weighted graph, with the symbols \(\left( {V\left( \Re \right),\;E\left( \Re \right),\;\varpi \left( {\varepsilon \kappa } \right)} \right)\). Here, the set of vertices and edges are denoted by \(V\left( \Re \right)\) and \(E\left( \Re \right)\) respectively, with \(\varpi \left( {\varepsilon \kappa } \right)\) signifying the edge weight of the graph \(\Re\). By adding the degrees of the edges \(\varepsilon\) and \(\kappa\), one may calculate the edge weight of \(\;\varpi \left( {\varepsilon \kappa } \right)\). The graph entropy based on edge weight is defined in Eq. (1).

$$ENT_{\varpi } \left( \Re \right) = - \sum\limits_{\varepsilon \kappa \in E(\Re )} {\frac{{\varpi \left( {\varepsilon \kappa } \right)}}{{\sum\limits_{\varepsilon \kappa \in E(\Re )} {\varpi \left( {\varepsilon \kappa } \right)} }}} \log \frac{{\varpi \left( {\varepsilon \kappa } \right)}}{{\sum\limits_{\varepsilon \kappa \in E(\Re )} {\varpi \left( {\varepsilon \kappa } \right)} }}$$
(1)

By using topological indices (see Table 2) in Eq. (1) we get following entropies which are shown in Table 2.

Table 2 Entropies of topological indices.
Full size table

After reviewing above mentioned literature, we found that computation of the graph of tin oxide (SnO2) for above mentioned indices and entropies are not discussed so far. Present research focuses on the QSPR analysis of the graph of tin oxide (SnO2) for above defined novel topological indices and the measures of entropy. Additionally, we use the linear regression model to correlate the indices and entropy. All of its results are displayed both graphically and mathematically using the appropriate line fit technique.

Results and discussions

In this section, we have computed different topological indices as well as their corresponding entropy measures by using the above-defined formulae.

Results for topological indices of tin oxide \(SnO_{2}\)

We thoroughly examined the computation of various degree-based topological indices in this section to give a detailed review of the structural characteristics of the considered structures. These indices were carefully calculated in order to evaluate the topological properties of molecular graphs, which in effect showed their connection patterns and, potentially, their influence on chemical behavior. Furthermore, we integrated graphical representations to our numerical comparisons in order to further improve our findings. Visualizing the patterns and changes in these indices made it easier to figure out how structural variations between different molecules affected their individual degree-based topological indices.

The graph of tin oxide (\(SnO_{2}\)) is presented in Fig. 1. We partitioned the edges of the graph according to the degree of the end vertices. All vertices having degrees according to edges connected with the respective vertices are computed as 1, 2, 3 and 6. Here we have six different types of edges whose end vertices have degree \((1,2)\),\((1,3)\), \((2,6)\), \((3,6)\),\((2,2)\) and \(\left( {2,\,3} \right)\). Symbolically represented by \(E_{1} (1,2)\),\(E_{2} (1,3)\), \(E_{3} (2,6)\), \(E_{4} (3,6)\),\(E_{5} (2,2)\) and \(E_{6} \left( {2,\,3} \right)\). Total number of edges computed of the type \(E_{1} (1,2)\),\(E_{2} (1,3)\), \(E_{3} (2,6)\), \(E_{4} (3,6)\),\(E_{5} (2,2)\) and \(E_{6} \left( {2,\,3} \right)\) are \(2pq + 2p\), \(2q + 2\), \(4pq\), \(2pq\), \(2p - 2\) and \(4p + q - 5\), respectively. All these results are summarized in Table 3.

Fig. 1
Fig. 1The alternative text for this image may have been generated using AI.
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Structure of tin oxide: a Unit cell of \(SnO_{2}\) and b Crystal structure \(SnO_{2}\)\(\left[ {3,\;3} \right]\).

Table 3 Edge partition of tin oxide (\(SnO_{2}\)).
Full size table

Theorem 1:

let \(SnO_{2}\) is the graph of tin oxide (\(SnO_{2}\)), then we have.

$$R_{1} (SnO_{2} ) = 88pq + 12q + 36p - 32$$
$$R_{ - 1} (SnO_{2} ) = 1.4444pq + 0.8339q + 2.1666p - 1.9999$$
$$R_{\frac{1}{2}} (SnO_{2} ) = 25.17pq + 16.6263p + 5.9158q - 12.881$$
$$R_{{\frac{ - 1}{2}}} (SnO_{2} ) = 3.0403pq + 4.0471p + 1.9552q - 1.4942$$
$$ABC(SnO_{2} ) = 5.4898pq + 5.6568p + 2.34q - 3.3168$$
$$GA(SnO_{2} ) = 7.2353pq + 7.8047p + 2.7117q - 5.1669$$
$$M_{1} (SnO_{2} ) = 56pq + 34p + 13q - 25$$
$$M_{2} (SnO_{2} ) = 88pq + 12q + 36p - 32$$
$$HM(SnO_{2} ) = 436pq + 150p + 57q - 125$$
$$F(SnO_{2} ) = 260pq + 78p + 33q - 61$$
$$AZI(SnO_{2} ) = 37.2243pq + 40p + 8.5q - 53.5$$

Proofs:

By using the edge partition given in Table 3 and formulae of the topological indices given in Table 1, we can prove the results as follows:

By using Randic index, if \(\alpha = 1\)

$$R_{1} (SnO_{2} ) = \sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi } ) \times (\psi (\Omega ))^{1} ;\;if\;\alpha = 1$$
$$R_{1} (SnO_{2} ) = (1 \times 2)(2pq + 2p) + (1 \times 3)(2q + 2) + (2 \times 6)(4pq) + (3 \times 6)(2pq) + (2 \times 2)(2p - 2) + (2 \times 3)(4p + q - 5)$$
$$R_{1} (SnO_{2} ) = 88pq + 12q + 36p - 32$$

By using Randic index, if \(\alpha = - 1\)

$$\begin{aligned} R_{ - 1} (SnO_{2} ) = & \sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi } ) \times (\psi (\Omega ))^{ - 1} ;\;if\,\alpha = - 1 \\ = & \frac{1}{(1 \times 2)}(2pq + 2p) + \frac{1}{(1 \times 3)}(2q + 2) + \frac{1}{(2 \times 6)}(4pq) + \frac{1}{(3 \times 6)}(2pq) \\ & + \frac{1}{(2 \times 2)}(2p - 2) + \frac{1}{(2 \times 3)}(4p + q - 5) \\ \end{aligned}$$
$$R_{ - 1} (SnO_{2} ) = 1.4444pq + 0.8339q + 2.1666p - 1.9999$$

By using Randic index, if \(\alpha = 1/2\)

$$\begin{aligned} R_{\frac{1}{2}} (SnO_{2} ) = & \sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi } ) \times (\psi (\Omega ))^{\frac{1}{2}} ;\;if\;\alpha = \frac{1}{2} \\ = & \sqrt {(1 \times 2)} (2pq + 2p) + \sqrt {(1 \times 3)} (2q + 2) + \sqrt {(2 \times 6)} (4pq) + \sqrt {(3 \times 6)} (2pq) \\ & + \sqrt {(2 \times 2)} (2p - 2) + \sqrt {(2 \times 3)} (4p + q - 5) \\ \end{aligned}$$
$$R_{\frac{1}{2}} (SnO_{2} ) = 25.17pq + 16.6263p + 5.9158q - 12.881$$

By using Randic index, if \(\alpha = - 1/2\).

$$\begin{aligned} R_{{\frac{ - 1}{2}}} (SnO_{2} ) = & \sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi } ) \times (\psi (\Omega ))^{{\frac{ - 1}{2}}} ;\,if\;\alpha = - \frac{1}{2} \\ \; = & \frac{1}{{\sqrt {(1 \times 2)} }}(2pq + 2p) + \frac{1}{{\sqrt {(1 \times 3)} }}(2q + 2) + \frac{1}{{\sqrt {(2 \times 6)} }}(4pq) + \sqrt {(3 \times 6)} (2pq) \\ & + \frac{1}{{\sqrt {(2 \times 2)} }}(2p - 2) + \frac{1}{{\sqrt {(2 \times 3)} }}(4p + q - 5), \\ \end{aligned}$$
$$R_{{\frac{ - 1}{2}}} (SnO_{2} ) = 3.0403pq + 4.0471p + 1.9552q - 1.4942.$$

By using Atom bond connectivity Index

$$\begin{aligned} ABC(G) = & \sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {\sqrt {\frac{(\psi (\xi ) + (\psi (\Omega )) - 2}{{(\psi (\xi ) \times (\psi (\Omega ))}}} } \\ = & \sqrt {\frac{(1 + 2 - 2)}{{(1 \times 2)}}} (2pq + 2p) + \sqrt {\frac{(1 + 3 - 2)}{{(1 \times 3)}}} (2q + 2) + \sqrt {\frac{(2 + 6 - 2)}{{(2 \times 6)}}} (4pq) + \sqrt {\frac{(3 + 6 - 2)}{{(3 \times 6)}}} (2pq) \\ & + \sqrt {\frac{(2 + 2 - 2)}{{(2 \times 2)}}} (2p - 2) + \sqrt {\frac{(2 + 3 - 2)}{{(2 \times 3)}}} (4p + q - 5) \\ \end{aligned}$$
$$ABC(SnO_{2} ) = 5.4898pq + 5.6568p + 2.34q - 3.3168$$

By using Geometric Arithmetic Index

$$\begin{aligned} GA(G) = & \sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {\frac{{2\sqrt {(\psi (\xi ) \times (\psi (\Omega ))} }}{(\psi (\xi ) + (\psi (\Omega ))}} \\ = & \frac{{2\sqrt {1 \times 2} }}{1 + 2}(2pq + 2p) + \frac{{2\sqrt {1 \times 3} }}{1 + 3}(2q + 2) + \frac{{2\sqrt {2 \times 6} }}{2 + 6}(4pq) + \frac{{2\sqrt {3 \times 6} }}{3 + 6}(2pq) \\ & + \frac{{2\sqrt {2 \times 2} }}{2 + 2}(2p - 2) + \frac{{2\sqrt {2 \times 3} }}{2 + 3}(4p + q - 5) \\ \end{aligned}$$
$$GA(SnO_{2} ) = 7.2353pq + 7.8047p + 2.7117q - 5.1669$$

By using first Zagreb Index

$$\begin{aligned} M_{1} (SnO_{2} ) = & \sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi } ) + (\psi (\Omega )) \\ = & (1 + 2)(2pq + 2p) + (1 + 3)(2q + 2) + (2 + 6)(4pq) + (3 + 6)(2pq) \\ & + (2 + 2)(2p - 2) + (2 + 3)(4p + q - 5) \\ \end{aligned}$$
$$\;\;\;\;\;\;\;\;\;\;\;M_{1} (SnO_{2} ) = 56pq + 34p + 13q - 25$$

By using second Zagreb Index

$$\begin{aligned} M_{2} (SnO_{2} ) = & \sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi } ) \times (\psi (\Omega )) \\ = & (1 \times 2)(2pq + 2p) + (1 \times 3)(2q + 2) + (2 \times 6)(4pq) + (3 \times 6)(2pq) \\ & + (2 \times 2)(2p - 2) + (2 \times 3)(4p + q - 5) \\ \end{aligned}$$
$$M_{2} (SnO_{2} ) = 88pq + 12q + 36p - 32$$

By using Hyper Zagreb Index

$$\begin{aligned} HM(G) = & \sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi } ) + (\psi (\Omega ))^{2} \\ = & (1 + 2)^{2} (2pq + 2p) + (1 + 3)^{2} (2q + 2) + (2 + 6)^{2} (4pq) + (3 + 6)^{2} (2pq) \\ & + (2 + 2)^{2} (2p - 2) + (2 + 3)^{2} (4p + q - 5) \\ \end{aligned}$$
$$HM(SnO_{2} ) = 436pq + 150p + 57q - 125$$

By using Forgotten Index

$$\begin{aligned} F(SnO_{2} ) = & \sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi } )^{2} + (\psi (\Omega )^{2} ) \\ = & (1^{2} + 2^{2} )(2pq + 2p) + (1^{2} + 3^{2} )(2q + 2) + (2^{2} + 6^{2} )(4pq) + (3^{2} + 6^{2} )(2pq) \\ & + (2^{2} + 2^{2} )(2p - 2) + (2^{2} + 3^{2} )(4p + q - 5) \\ \end{aligned}$$
$$F(SnO_{2} ) = 260pq + 78p + 33q - 61$$

By using Augmented Zagreb Index

$$\begin{aligned} AZI(SnO_{2} ) = & \sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {\left[ {\frac{\psi (\xi ) \times (\psi (\Omega )}{{\psi (\xi ) + (\psi (\Omega ) - 2}}} \right]}^{23} \\ = & \left( {\frac{1 \times 2}{{1 + 2 - 2}}} \right)^{3} (2pq + 2p) + \left( {\frac{1 \times 3}{{1 + 3 - 2}}} \right)^{3} (2q + 2) + \left( {\frac{2 \times 6}{{2 + 6 - 2}}} \right)^{3} (4pq) + \left( {\frac{3 \times 6}{{3 + 6 - 2}}} \right)^{3} (2pq) \\ & + \left( {\frac{2 \times 2}{{2 + 2 - 2}}} \right)^{3} (2p - 2) + \left( {\frac{2 \times 3}{{2 + 3 - 2}}} \right)^{3} (4p + q - 5) \\ \end{aligned}$$
$$AZI(SnO_{2} ) = 82.005831pq + 64p + 14.5q - 49.25$$

Computation of entropies

Let \(\Re\) be an edge-weighted graph, with the symbols \(\left( {V\left( \Re \right),\;E\left( \Re \right),\;\varpi \left( {\varepsilon \kappa } \right)} \right)\). Here, the set of vertices and edges are denoted by \(V\left( \Re \right)\) and \(E\left( \Re \right)\) respectively, with \(\varpi \left( {\varepsilon \kappa } \right)\) signifying the edge weight of the graph \(\Re\). By adding the degrees of the edges \(\varepsilon\) and \(\kappa\), one may calculate the edge weight of \(\;\varpi \left( {\varepsilon \kappa } \right)\). The graph entropy based on edge weight is defined in Eq. (1).

$$ENT_{\varpi } \left( \Re \right) = - \sum\limits_{\varepsilon \kappa \in E(\Re )} {\frac{{\varpi \left( {\varepsilon \kappa } \right)}}{{\sum\limits_{\varepsilon \kappa \in E(\Re )} {\varpi \left( {\varepsilon \kappa } \right)} }}} \log \frac{{\varpi \left( {\varepsilon \kappa } \right)}}{{\sum\limits_{\varepsilon \kappa \in E(\Re )} {\varpi \left( {\varepsilon \kappa } \right)} }}$$
(1)

By using topological indices (see Table 1) in Eq. (1) we get the entropies shown in Table 2.

Let \(SnO_{2}\) is the graph of tin oxide then the entropies of the considered topological indices can be computed as follows:

By using the edge partition given in Table 3 and formulae of the entropies given in Table 2, we can easily obtain the following results.

By using Randic entropy, if \(\alpha = 1\)

$$ENT_{{R_{1} (SnO_{2} )}} = \log R_{1} (SnO_{2} ) - \frac{1}{{R_{1} (SnO_{2} )}}\sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi } ) \times (\psi (\Omega ))^{1} \log (\psi (\xi ) \times (\psi (\Omega ))^{1} ;\;if\;\alpha = 1$$
$$\begin{aligned} ENT_{{R_{1} (SnO_{2} )}} = & \log (88pq + 12q + 36p - 32) - \frac{(2)(2pq + 2p)\log (2)}{{(88pq + 12q + 36p - 32)}} - \frac{(3)(2q + 2)\log (3)}{{(88pq + 12q + 36p - 32)}} \\ & - \frac{(12)(4pq)\log (12)}{{(88pq + 12q + 36p - 32)}} - \frac{(18)(2pq)\log (18)}{{(88pq + 12q + 36p - 32)}} - \frac{(4)(2p - 2)\log (4)}{{(88pq + 12q + 36p - 32)}} \\ & - \frac{(6)(4p + q - 5)\log (6)}{{(88pq + 12q + 36p - 32)}} \\ \end{aligned}$$

By using Randic entropy, if \(\alpha = - 1\)

$$ENT_{{R_{ - 1} (SnO_{2} )}} = \log R_{ - 1} (SnO_{2} ) - \frac{1}{{R_{ - 1} (SnO_{2} )}}\sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi } ) \times (\psi (\Omega ))^{ - 1} \log (\psi (\xi ) \times (\psi (\Omega ))^{ - 1} ;\;if\;\alpha = - 1$$

By using Randic entropy, if \(\alpha = 1/2\)

$$ENT_{{R_{\frac{1}{2}} (SnO_{2} )}} = \log R_{ - 1} (SnO_{2} ) - \frac{1}{{R_{ - 1} (SnO_{2} )}}\sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi } ) \times (\psi (\Omega ))^{\frac{1}{2}} \log (\psi (\xi ) \times (\psi (\Omega ))^{\frac{1}{2}} ;\;for\;\alpha = \frac{1}{2}$$
$$\begin{aligned} ENT_{{R_{\frac{1}{2}} }} (SnO_{2} ) = & \log (25.17pq + 16.6263p + 5.9158q - 12.881) - \frac{{\sqrt {(2)} (2pq + 2p)\log \sqrt {(2)} }}{(25.17pq + 16.6263p + 5.9158q - 12.881)} \\ & - \frac{{\sqrt {(3)} (2q + 2)\log \sqrt {(3)} }}{(25.17pq + 16.6263p + 5.9158q - 12.881)} - \frac{{\sqrt {(12)} (4pq)\log \sqrt {(12)} }}{(25.17pq + 16.6263p + 5.9158q - 12.881)} \\ & - \frac{{\sqrt {(18)} (2pq)\log \sqrt {(18)} }}{(25.17pq + 16.6263p + 5.9158q - 12.881)} - \frac{{\sqrt {(4)} (2p - 2)\log \sqrt {(4)} }}{(25.17pq + 16.6263p + 5.9158q - 12.881)} \\ & - \frac{{\sqrt {(6)} (4p + q - 5)\log \sqrt {(6)} }}{(25.17pq + 16.6263p + 5.9158q - 12.881)} \\ \end{aligned}$$

By using Randic entropy, if \(\alpha = - 1/2\)

$$ENT_{{R_{{\frac{ - 1}{2}}} (SnO_{2} )}} = \log R_{{\frac{ - 1}{2}}} (SnO_{2} ) - \frac{1}{{R_{{\frac{ - 1}{2}}} (SnO_{2} )}}\sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi } ) \times (\psi (\Omega ))^{{ - \frac{1}{2}}} \log (\psi (\xi ) \times (\psi (\Omega ))^{{ - \frac{1}{2}}} ;\;for\;\alpha = \frac{1}{2}$$
$$\begin{aligned} ENTR_{{_{{ - \frac{1}{2}}} }} (SnO_{2} ) = & \log (3.0403pq + 4.0471p + 1.9552q - 1.4942) - \frac{{\left( {\frac{1}{{\sqrt {(2)} }}} \right)(2pq + 2p)\log \left( {\frac{1}{{\sqrt {(2)} }}} \right)}}{(3.0403pq + 4.0471p + 1.9552q - 1.4942)} \\ & - \frac{{\left( {\frac{1}{{\sqrt {(3)} }}} \right)(2q + 2)\log \left( {\frac{1}{{\sqrt {(3)} }}} \right)}}{(3.0403pq + 4.0471p + 1.9552q - 1.4942)} - \frac{{\left( {\frac{1}{{\sqrt {(12)} }}} \right)(4pq)\log \left( {\frac{1}{{\sqrt {(12)} }}} \right)}}{(3.0403pq + 4.0471p + 1.9552q - 1.4942)} \\ & - \frac{{\left( {\frac{1}{{\sqrt {(18)} }}} \right)(2pq)\log \left( {\frac{1}{{\sqrt {(18)} }}} \right)}}{(3.0403pq + 4.0471p + 1.9552q - 1.4942)} - \frac{{\left( {\frac{1}{{\sqrt {(4)} }}} \right)(2p - 2)\log \left( {\frac{1}{{\sqrt {(4)} }}} \right)}}{(3.0403pq + 4.0471p + 1.9552q - 1.4942)} \\ & - \frac{{\left( {\frac{1}{{\sqrt {(6)} }}} \right)(4p + q - 5)\log \left( {\frac{1}{{\sqrt {(6)} }}} \right)}}{(3.0403pq + 4.0471p + 1.9552q - 1.4942)} \\ \\ \end{aligned}$$

By using Atom bond connectivity entropy

$$ENT_{{ABC(SnO_{2} )}} = \log R_{{\frac{ - 1}{2}}} (SnO_{2} ) - \frac{1}{{R_{{\frac{ - 1}{2}}} (SnO_{2} )}}\sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {\sqrt {\frac{(\psi (\xi ) + (\psi (\Omega )) - 2}{{(\psi (\xi ) \times (\psi (\Omega ))}}} } \log \sqrt {\frac{(\psi (\xi ) + (\psi (\Omega )) - 2}{{(\psi (\xi ) \times (\psi (\Omega ))}}}$$
$$\begin{aligned} ENT_{{ABC(SnO_{2} )}} = & \log (5.4898pq + 5.6568p + 2.34q - 3.3168) - \frac{{\left( {\sqrt{\frac{1}{2}} } \right)(2pq + 2p)\log \left( {\sqrt{\frac{1}{2}} } \right)}}{(5.4898pq + 5.6568p + 2.34q - 3.3168)} \\ & - \frac{{\left( {\sqrt{\frac{4}{3}} } \right)(2q + 2)\log \left( {\sqrt{\frac{4}{3}} } \right)}}{(5.4898pq + 5.6568p + 2.34q - 3.3168)} - \frac{{\left( {\sqrt{\frac{6}{12}} } \right)(4pq)\log \left( {\sqrt{\frac{6}{12}} } \right)}}{(5.4898pq + 5.6568p + 2.34q - 3.3168)} \\ & - \frac{{\left( {\sqrt{\frac{7}{18}} } \right)(2pq)\log \left( {\sqrt{\frac{7}{18}} } \right)}}{(5.4898pq + 5.6568p + 2.34q - 3.3168)} - \frac{{\left( {\sqrt{\frac{2}{4}} } \right)(2p - 2)\log \left( {\sqrt{\frac{2}{4}} } \right)}}{(5.4898pq + 5.6568p + 2.34q - 3.3168)} \\ & - \frac{{\left( {\sqrt{\frac{3}{6}} } \right)(4p + q - 5)\log \left( {\sqrt{\frac{3}{6}} } \right)}}{(5.4898pq + 5.6568p + 2.34q - 3.3168)} \\ \end{aligned}$$

By using Geometric arithmetic entropy

$$ENT_{{GA(SnO_{2} )}} = \log GA(SnO_{2} ) - \frac{1}{{GA(SnO_{2} )}}\sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {\frac{{2\sqrt {(\psi (\xi ) \times (\psi (\Omega ))} }}{(\psi (\xi ) + (\psi (\Omega ))}} \log \frac{{2\sqrt {(\psi (\xi ) \times (\psi (\Omega ))} }}{(\psi (\xi ) + (\psi (\Omega ))}$$
$$\begin{aligned} ENT_{{GA(SnO_{2} )}} = & \log (7.2353pq + 7.8047p + 2.7117q - 5.1669) - \frac{{\left( {\frac{2\sqrt 2 }{3}} \right)(2pq + 2p)\log \left( {\frac{2\sqrt 3 }{4}} \right)}}{(7.2353pq + 7.8047p + 2.7117q - 5.1669)} \\ & - \frac{{\left( {\frac{{2\sqrt {1 \times 2} }}{1 + 2}} \right)(2q + 2)\log \left( {\frac{{2\sqrt {1 \times 2} }}{1 + 2}} \right)}}{(7.2353pq + 7.8047p + 2.7117q - 5.1669)} - \frac{{\left( {\frac{{2\sqrt {12} }}{8}} \right)(4pq)\log \left( {\frac{{2\sqrt {18} }}{9}} \right)}}{(7.2353pq + 7.8047p + 2.7117q - 5.1669)} \\ & - \frac{{\left( {\frac{2\sqrt 4 }{4}} \right)(2pq)\log \left( {\frac{2\sqrt 4 }{4}} \right)}}{(7.2353pq + 7.8047p + 2.7117q - 5.1669)} - \frac{{\left( {\frac{{2\sqrt {1 \times 2} }}{1 + 2}} \right)(2p - 2)\log \left( {\frac{{2\sqrt {1 \times 2} }}{1 + 2}} \right)}}{(7.2353pq + 7.8047p + 2.7117q - 5.1669)} \\ & - \frac{{\left( {\frac{2\sqrt 6 }{5}} \right)(4p + q - 5)\log \left( {\frac{2\sqrt 6 }{5}} \right)}}{(7.2353pq + 7.8047p + 2.7117q - 5.1669)} \\ \end{aligned}$$

By using first Zagreb entropy

$$\begin{aligned} ENT_{{M_{1} (SnO_{2} )}} = & \log (56pq + 34p + 13q - 25) - \frac{(3)(2pq + 2p)\log (3)}{{(56pq + 34p + 13q - 25)}} - \frac{(4)(2q + 2)\log (4)}{{(56pq + 34p + 13q - 25)}} \\ & - \frac{(8)(4pq)\log (8)}{{(56pq + 34p + 13q - 25)}} - \frac{(9)(2pq)\log (9)}{{(56pq + 34p + 13q - 25)}} - \frac{(4)(2p - 2)\log (4)}{{(56pq + 34p + 13q - 25)}} \\ & - \frac{(5)(4p + q - 5)\log (5)}{{(56pq + 34p + 13q - 25)}} \\ \end{aligned}$$

By using second Zagreb entropy

$$ENT_{{M_{2} (SnO_{2} )}} = \log GA(SnO_{2} ) - \frac{1}{{GA(SnO_{2} )}}\sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi ) \times (\psi (\Omega ))} \log (\psi (\xi ) \times (\psi (\Omega ))$$
$$\begin{aligned} ENT_{{M_{2} (SnO_{2} )}} = & \log (88pq + 12q + 36p - 32) - \frac{(2)(2pq + 2p)\log (2)}{{(88pq + 12q + 36p - 32)}} - \frac{(3)(2q + 2)\log (3)}{{(88pq + 12q + 36p - 32)}} \\ & - \frac{(12)(4pq)\log (12)}{{(88pq + 12q + 36p - 32)}} - \frac{(18)(2pq)\log (18)}{{(88pq + 12q + 36p - 32)}} - \frac{(4)(2p - 2)\log (4)}{{(88pq + 12q + 36p - 32)}} \\ & - \frac{(6)(4p + q - 5)\log (6)}{{(88pq + 12q + 36p - 32)}} \\ \end{aligned}$$

By using Hyper Zagreb entropy

$$ENT_{{HM(SnO_{2} )}} = \log HM(SnO_{2} ) - \frac{1}{{HM(SnO_{2} )}}\sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi ) + (\psi (\Omega ))^{2} } \log (\psi (\xi ) + (\psi (\Omega ))^{2}$$
$$\begin{aligned} ENT_{{HM(SnO_{2} )}} = & \log (436pq + 150p + 57q - 125) - \frac{(9)(2pq + 2p)\log (9)}{{(436pq + 150p + 57q - 125)}} - \frac{(16)(2q + 2)\log (16)}{{(436pq + 150p + 57q - 125)}} \\ & - \frac{(64)(4pq)\log (64)}{{(436pq + 150p + 57q - 125)}} - \frac{(81)(2pq)\log (81)}{{(436pq + 150p + 57q - 125)}} - \frac{(16)(2p - 2)\log (16)}{{(436pq + 150p + 57q - 125)}} \\ & - \frac{(25)(4p + q - 5)\log (25)}{{(436pq + 150p + 57q - 125)}} \\ \end{aligned}$$

By using forgotten entropy

$$ENT_{{F(SnO_{2} )}} = \log F(SnO_{2} ) - \frac{1}{{F(SnO_{2} )}}\sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {(\psi (\xi )^{2} + (\psi (\Omega )^{2} )} \log (\psi (\xi )^{2} + (\psi (\Omega )^{2} )$$
$$\begin{aligned} ENT_{{F(SnO_{2} )}} = & \log (260pq + 78p + 33q - 61) - \frac{(5)(2pq + 2p)\log (5)}{{(260pq + 78p + 33q - 61)}} \\ & - \frac{(10)(2q + 2)\log (10)}{{(260pq + 78p + 33q - 61)}} - \frac{(40)(4pq)\log (40)}{{(260pq + 78p + 33q - 61)}} \\ & - \frac{(45)(2pq)\log (45)}{{(260pq + 78p + 33q - 61)}} - \frac{(8)(2p - 2)\log (8)}{{(260pq + 78p + 33q - 61)}} \\ & - \frac{(13)(4p + q - 5)\log (13)}{{(260pq + 78p + 33q - 61)}} \\ \end{aligned}$$

By using Augmented Zagreb entropy

$$ENT_{{AZI(SnO_{2} )}} = \log AZI(SnO_{2} ) - \frac{1}{{AZI(SnO_{2} )}}\sum\limits_{{\xi \Omega \in E(SnO_{2} )}} {\left[ {\frac{\psi (\xi ) \times (\psi (\Omega )}{{\psi (\xi ) + (\psi (\Omega ) - 2}}} \right]}^{3} \log \left[ {\frac{\psi (\xi ) \times (\psi (\Omega )}{{\psi (\xi ) + (\psi (\Omega ) - 2}}} \right]^{3}$$
$$\begin{aligned} ENT_{{AZI(SnO_{2} )}} = & \log (82.005831pq + 64p + 14.5q - 49.25) - \frac{(2)(2pq + 2p)\log (2)}{{(82.005831pq + 64p + 14.5q - 49.25)}} \\ & - \frac{{\left( \frac{9}{4} \right)(2q + 2)\log \left( \frac{9}{4} \right)}}{(82.005831pq + 64p + 14.5q - 49.25)} - \frac{{\left( {\frac{144}{{36}}} \right)(4pq)\log \left( {\frac{144}{{36}}} \right)}}{(82.005831pq + 64p + 14.5q - 49.25)} \\ & - \frac{{\left( {\frac{324}{{49}}} \right)(2pq)\log \left( {\frac{324}{{49}}} \right)}}{(82.005831pq + 64p + 14.5q - 49.25)} - \frac{(4)(2p - 2)\log (4)}{{(82.005831pq + 64p + 14.5q - 49.25)}} \\ & - \frac{(4)(4p + q - 5)\log (4)}{{(82.005831pq + 64p + 14.5q - 49.25)}} \\ \end{aligned}$$

Quantification and visualization of results

Comparison of results for topological indices

Tables 4, 5, 6 and Figs. 2, 3, 4 provide a comprehensive numerical and graphical comparison of several indices as the parameters p and q increased. The findings clearly reveal a positive increase by indicating that all indices increase when p and q values increase. But the rates of rise of the indices are different. Specifically, the graphical representations show that the \(HM(SnO_{2} )\) index is increasing more rapidly than the other indices. This indicates that \(HM(SnO_{2} )\) is more responsive to changes in p and q, which may be due to the underlying computing method. The significant increase in \(HM(SnO_{2} )\) indicates that it has the potential to be a very responsive measure, which might be particularly useful in applications requiring fast or highly accurate detection of changes.

Table 4 Comparison of topological indices for tin oxide (\(SnO_{2}\)).
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Table 5 Comparison of topological indices for tin oxide (\(SnO_{2}\)).
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Table 6 Comparison of topological indices for tin oxide (\(SnO_{2}\)).
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Fig. 2
Fig. 2The alternative text for this image may have been generated using AI.
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Graphical comparison of topological indices \(R_{1} (SnO_{2} )\),\(R_{ - 1} (SnO_{2} )\),\(R_{1/2} (SnO_{2} )\) and \(R_{ - 1/2} (SnO_{2} )\).

Fig. 3
Fig. 3The alternative text for this image may have been generated using AI.
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Graphical comparison of topological indices \(ABC(SnO_{2} )\), \(GA(SnO_{2} )\), \(M_{1} (SnO_{2} )\) and \(M_{2} (SnO_{2} )\).

Fig. 4
Fig. 4The alternative text for this image may have been generated using AI.
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Graphical comparison of topological indices \(HM(SnO_{2} )\), \(F(SnO_{2} )\) and \(AZI(SnO_{2} )\).

Comparison of results for entropies

Tables 7, 8, 9 and Figs. 5, 6, 7 present a comprehensive numerical and graphical comparison of various entropies as the parameters p and q are increased. By demonstrating that all indices rise with higher values of p and q, the results clearly establish a positive correlation. However, the rates of rise of the indices are not the same. The \(HM(SnO_{2} )\) entropy is rising more quickly than the other entropy, according to the graphical representation in particular. It indicates that the reason \(ENT_{{HM(SnO_{2} )}}\) is more sensitive to changes in p and q could be related to its underlying computation method. The quick increases in \(ENT_{{HM(SnO_{2} )}}\) shows its potential as very responsive, which could be very useful in applications that need fast or extremely sensitive detection of changes.

Table 7 Comparison of Entropies for tin oxide (\(SnO_{2}\)).
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Table 8 Comparison of Entropies for tin oxide (\(SnO_{2}\)).
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Table 9 Comparison of Entropies for tin oxide (\(SnO_{2}\)).
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Fig. 5
Fig. 5The alternative text for this image may have been generated using AI.
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Graphical comparison of entropy of \(ENT_{{R_{1} (SnO_{2} )}}\), \(ENT_{{R_{ - 1} (SnO_{2} )}}\), \(ENT_{{R_{1/2} (SnO_{2} )}}\) and \(ENT_{{R_{ - 1/2} (SnO_{2} )}}\).

Fig. 6
Fig. 6The alternative text for this image may have been generated using AI.
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Graphical comparison of entropy of \(ENT_{{ABC(SnO_{2} )}}\), \(ENT_{{GA(SnO_{2} )}}\), \(ENT_{{M_{1} (SnO_{2} )}}\) and \(ENT_{{M_{2} (SnO_{2} )}}\).

Fig. 7
Fig. 7The alternative text for this image may have been generated using AI.
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Graphical comparison of entropy of \(ENT_{{HM(SnO_{2} )}}\), \(ENT_{{F(SnO_{2} )}}\) and \(ENT_{{AZI(SnO_{2} )}}\).

Relationship between topological indices and entropies

We use the following linear regression model to investigate the relationship between topological indices and their corresponding entropies.

$$ENT = a\left[ {TI} \right] + b$$
(2)

where ENT be the entropy measure, TI be the topological indices, a be the slope and b be the y intercept. We establish the relationship between the entropy measure and the degree-based topological indices using the aforementioned equation (Eq. 2). The correlation coefficient R is measure of degree that predict change in dependent variable with respect to independent variable. We determined each degree-based entropy for the different values of p and q for the tin oxide structure. In Figs. 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, the line fitting between degree-based indices and entropy measure is displayed. Incorporating a curve into data allows for the investigation of the relationship between many kinds of variables. We investigated the relationship between entropy formation and several indices using this technique. By adjusting several basic variables, the correlation between entropy and all indices was estimated using the linear curve fitting method. The linear regression approach, standard error estimation, \(R\), \(R^{2}\), and coefficient are the accuracy measurements used in this investigation. Table 10 presents the correlation coefficient values for each topological index against entropy.

Fig. 8
Fig. 8The alternative text for this image may have been generated using AI.
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Curve fitting between \(_{{R_{1} (SnO_{2} )}}\) and \(ENT_{{R_{1} (SnO_{2} )}}\).

Fig. 9
Fig. 9The alternative text for this image may have been generated using AI.
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Curve fitting between \(_{{R_{ - 1} (SnO_{2} )}}\) and \(ENT_{{R_{ - 1} (SnO_{2} )}}\).

Fig. 10
Fig. 10The alternative text for this image may have been generated using AI.
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Curve fitting between \(_{{R_{1/2} (SnO_{2} )}}\) and \(ENT_{{R_{1/2} (SnO_{2} )}}\).

Fig. 11
Fig. 11The alternative text for this image may have been generated using AI.
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Curve fitting between \(R_{ - 1/2} (SnO_{2} )\) and \(ENT_{{R_{ - 1/2} (SnO_{2} )}}\).

Fig. 12
Fig. 12The alternative text for this image may have been generated using AI.
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Curve fitting between \(M_{2} (SnO_{2} )\) and \(ENT_{{M_{2} (SnO_{2} )}}\).

Fig. 13
Fig. 13The alternative text for this image may have been generated using AI.
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Curve fitting between \(M_{1} (SnO_{2} )\) and \(ENT_{{M_{1} (SnO_{2} )}}\).

Fig. 14
Fig. 14The alternative text for this image may have been generated using AI.
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Curve fitting between \(GA(SnO_{2} )\) and \(ENT_{{GA(SnO_{2} )}}\).

Fig. 15
Fig. 15The alternative text for this image may have been generated using AI.
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Curve fitting between \(ABC(SnO_{2} )\) and \(ENT_{{ABC(SnO_{2} )}}\).

Fig. 16
Fig. 16The alternative text for this image may have been generated using AI.
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Curve fitting between \(AZI(SnO_{2} )\) and \(ENT_{{AZI(SnO_{2} )}}\).

Fig. 17
Fig. 17The alternative text for this image may have been generated using AI.
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Curve fitting between \(F(SnO_{2} )\) and \(ENT_{{F(SnO_{2} )}}\).

Fig. 18
Fig. 18The alternative text for this image may have been generated using AI.
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Curve fitting between \(HM(SnO_{2} )\) and \(ENT_{{HM(SnO_{2} )}}\).

Table10 Statistical parameters for linear regression model.
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When we compared the value of \(ABC(SnO_{2} )\) to the value of other topological indices, the values of \(R\) and \(R^{2}\) in Table 10 and Fig. 15 are maximum, while the value of SE is minimal. So, the best predictor is \(ABC(SnO_{2} )\). The following linear regression models are obtained for each index and entropy:

$$ENT_{{R_{1} (SnO_{2} )}} = 0.0002409823\left[ {R_{1} (SnO_{2} )} \right] + 1.4900085298$$
$$ENT_{{R_{ - 1} (SnO_{2} )}} = 0.0092371366\left[ {R_{ - 1} (SnO_{2} )} \right] + 1.56410708647$$
$$ENT_{{R_{\frac{1}{2}} (SnO_{2} )}} = 0.00077444537\left[ {R_{\frac{1}{2}} (SnO_{2} )} \right] + 1.5782860469$$
$$ENT_{{R_{{\frac{ - 1}{2}}} (SnO_{2} )}} = 0.00986245096\left[ {R_{{\frac{ - 1}{2}}} (SnO_{2} )} \right] + 1.63332471835$$
$$ENT_{{ABC(SnO_{2} )}} = 0.0033520492\left[ {ABC(SnO_{2} )} \right] + 1.556309914$$
$$ENT_{{GA(SnO_{2} )}} = 0.00260961815\left[ {GA(SnO_{2} )} \right] + 1.58110014738$$
$$ENT_{{M_{1} (SnO_{2} )}} = 0.00036513092\left[ {M_{1} (SnO_{2} )} \right] + 1.709767996$$
$$ENT_{{M_{2} (SnO_{2} )}} = 0.0002426289\left[ {M_{2} (SnO_{2} )} \right] + 1.5126258796$$
$$ENT_{{HM(SnO_{2} )}} = 0.000048258787\left[ {HM(SnO_{2} )} \right] + 1.5028938395$$
$$ENT_{{F(SnO_{2} )}} = 0.000081253345\left[ {F(SnO_{2} )} \right] + 1.50081910045$$
$$ENT_{{AZI(SnO_{2} )}} = 0.00025106391\left[ {AZI(SnO_{2} )} \right] + 2.4568021004$$

Table 11 shows the predicted entropy obtained from linear regression models (see Eq. 2), whereas Tables 4, 5, 6 show the entropy values obtained from Eq. 1.

Table 11 Predicted Entropies.
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Conclusion

In this study, we developed and computed some degree based topological indices and their corresponding entropies for the molecular structure of tin oxide in order to provide more information on the structural characteristics of these systems. Our computations of entropy measures, which were based on these indices, showed the intricate relationship between structural complexity and information content. We used a linear regression model in an attempt to provide additional insight into the connections. The results demonstrated the feasibility of the model and emphasized its practical significance. The visualization of results demonstrated a strong fit between the regression curve and the scattered data points, validating our findings. These insights have significant implications for nanotechnology, where understanding the structure–property relationships at the molecular level is crucial for designing novel nanomaterials with tailored functionalities. In catalysis, the correlation between structural complexity and entropy can aid in predicting active sites and optimizing catalyst design for enhanced efficiency. Moreover, the topological indices and entropy measures developed in this study offer valuable tools for materials prediction, enabling the identification of promising molecular configurations with desirable stability and reactivity. The compatibility of model with the data demonstrates the validity of our analytical framework and its capacity to describe the complicated dynamics of complex systems. By promoting a deeper comprehension of the relationships between structural characteristics, information entropy, and regression modeling, this research provides a framework for further studies in complex system analysis.