Introduction

According to graph theory, a graph is made up of vertices V(G), or groups of objects, connected by edges E(G), or straight lines1,2. We deal with simple graphs, which means that they don’t have self-loops or multiple, directed, or weighted edges. \(\eta _{\alpha }\) represents the degree of a vertex \(\alpha\), which is the number of edges connected to it.

The physical, chemical, and topological characteristics of a graph are displayed by the topological index’s numerical value3,4. Chemical graph theory allows us to model the mathematical behavior of chemical networks. It has to do with using graph theory to a complex chemical issue. In the discipline of chemical chemistry, this idea is extremely important. Information science, mathematics, and chemistry are combined to make chem-informatics5,6. A mathematical technique known as a topological graph index, or molecular descriptor, may be applied to any graph that represents a molecule structure. This index may be used to further study certain physicochemical aspects of a molecule and assess mathematical values. In the last 20 years, a great deal of topological indices, or numerical graph invariants, have been created and applied to correlation analysis in environmental chemistry, toxicology, theoretical chemistry, and pharmacology. To what degree these indices are associated with one another, however, has not been the subject of any comprehensive investigation21.

Shanmukha, et al.7,8 examined the curved regression models and chemical application of the vertex-degree-based topological index. Zhang et al.9,10 examined the degree-based topological indexes of certain anti-cancer treatments using curve fitting models. Rasheed et al.11 discussed the octane isomer characteristics using new degree-based topological indices. Rai et al.13 analyzed the comparison of M-polynomial based topological indices between subdivisions and poly Hex-derived networks. Abirami et al.15,16 computed degree-based topological indices for ruthenium bipyridine’s complex structure. Kirana et al.18 discussed the Quinolone antibiotics’ different degree-based topological indices using QSPR and curvilinear regression. Tousi et al.20,35 computed the degree-based topological indices of Titanium dioxide nanotubes’ molecular graph and line graph. Yu et al.22,23 used the logarithmic regression model’s application to the topological indices and entropy metrics of the beryllunite network. Zaman et al.24 regression models and degree-based topological indicators were utilised in the QSPR study of a few new medications used to treat blood cancer.

In this paper, we discuss degree-based topological indices. In 1972, the \({M}_1(G), {M}_2(G)\) indices were presented by Gutman25 and defined in Eq.1 and Eq.2 as follows:

$$\begin{aligned} & {M}_1(G)=\sum _{\alpha \beta \in {E}(G)}(\eta _{\alpha }+\eta _{\beta }). \end{aligned}$$
(1)
$$\begin{aligned} & \quad {M}_2(G)=\sum _{\alpha \beta \in {E}(G)}(\eta _{\alpha }\times \eta _{\beta }). \end{aligned}$$
(2)

Tang et al.26 defined the third, fourth, and fifth Zagreb indices shown in Eq. 3 and Eq. 4.

$$\begin{aligned} & {M}_4(G)=\sum _{\alpha \beta \in {E}(G)}\eta _{\alpha }(\eta _{\alpha }+\eta _{\beta }). \end{aligned}$$
(3)
$$\begin{aligned} & {M}_5(G)=\sum _{\alpha \beta \in {E}(G)}\eta _{\beta }(\eta _{\alpha }+\eta _{\beta }). \end{aligned}$$
(4)

In 2016, Gao27 discussed their work and defined redefined Zagreb indices as shown in Eq. 5, Eq. 6, and Eq. 7:

$$\begin{aligned} & {ReZG}_1(G)=\sum _{\alpha \beta \in {E}(G)}{\frac{\eta _{\alpha }+\eta _{\beta }}{\eta _{\alpha }\times \eta _{\beta }}}. \end{aligned}$$
(5)
$$\begin{aligned} & \quad {ReZG}_2(G)=\sum _{\alpha \beta \in {E}(G)}{\frac{\eta _{\alpha }\times \eta _{\beta }}{\eta _{\alpha }+\eta _{\beta }}}. \end{aligned}$$
(6)
$$\begin{aligned} & {ReZG}_3(G)=\sum _{\alpha \beta \in {E}(G)}{(\eta _{\alpha }\times \eta _{\beta })}{(\eta _{\alpha }+\eta _{\beta })}. \end{aligned}$$
(7)

The entropy of a probability quantifies the degree of uncertainty in a system. The statistical methods serve as a solid foundation for this idea. It is mostly used for chemical structures and the graphs that go with them. It also offers details on chemical topologies and graph structure. It was initially employed as a concept in 1955. Entropy has uses in a wide range of technological and scientific domains. This determines both intrinsic and extrinsic entries. Mondal, S., and Das, K. C.12,29 discussed the degree-based graph entropy in Structure-Property Modeling. Emadi Kouchak et al.14,30 danalyzed the structural irregularity in networks using graph entropies-graph energies indices. Jacob et al.31 computed Entropy measurements and topological characterisation of tetragonal zeolite merlinoites. Ishfaq32 discussed the diamond structure entropies and topological indices. Networks as information functionalities are investigated using the concept of degree power. The concept of entropy for various topological indices is proposed by the writers. The basis discussed in28 is the entropy of probability distributions. The information entropy is defined as:

$$\begin{aligned} ENT_{\psi }(G)=-\sum \limits _{i=1}^{q}N_{i}\frac{G(\alpha _{i}\beta _{i})}{TI}\log \frac{G(\alpha _{i}\beta _{i})}{TI} =\log (TI)-\frac{1}{TI}\sum \limits _{i=1}^{q}N_{i}G(\alpha _{i}\beta _{i})\log {G(\alpha _{i}\beta _{i})}. \end{aligned}$$
(8)

  • First Zagreb Entropy

If \(G(\alpha \beta )=(\eta _{\alpha }+\eta _{\beta })\), then

$$\begin{aligned} \sum \limits _{\alpha \beta \in E }G(\alpha \beta )=\sum \limits _{\alpha \beta \in E }(\eta _{\alpha }+\eta _{\beta })=M_{1} \end{aligned}$$

Now Eq. (8) represents the first Zagreb entropy.

$$\begin{aligned} ENT_{M_{1}}(G)=\log \big (M_{1}\big )-\frac{1}{\big (M_{1}\big )} \sum \limits _{i=1}^{q}\sum \limits _{\alpha \beta \in E_{i}} [(\eta _{\alpha }+\eta _{\beta })]\log \big [(\eta _{\alpha }+\eta _{\beta })\big ]. \end{aligned}$$
(9)

  • Second Zagreb Entropy

If \(G(\alpha \beta )=(\eta _{\alpha }\times \eta _{\beta })\), then

$$\begin{aligned} \sum \limits _{\alpha \beta \in E }G(\alpha \beta )=\sum \limits _{\alpha \beta \in E }(\eta _{\alpha }\times \eta _{\beta })=M_{2} \end{aligned}$$

Now Equation (8) represents the second Zagreb entropy.

$$\begin{aligned} ENT_{M_{2}}(G)=\log \big (M_{2}\big )-\frac{1}{\big (M_{2}\big )} \sum \limits _{i=1}^{q}\sum \limits _{\alpha \beta \in E_{i}}[(\eta _{\alpha }\times \eta _{\beta })]\log \big [(\eta _{\alpha }\times \eta _{\beta })\big ]. \end{aligned}$$
(10)

  • Fourth Zagreb Entropy

If \(G(\alpha \beta )=\eta _{\alpha }(\eta _{\alpha }+\eta _{\beta })\), then

$$\begin{aligned} \sum \limits _{\alpha \beta \in E }G(\alpha \beta )=\sum \limits _{\alpha \beta \in E }\eta _{\alpha }(\eta _{\alpha }+\eta _{\beta })=M_{4} \end{aligned}$$

Now Equation (8) represents the fourth Zagreb entropy.

$$\begin{aligned} ENT_{M_{4}}(G)=\log \big (M_{4}\big )-\frac{1}{\big (M_{4}\big )} \sum \limits _{i=1}^{q}\sum \limits _{\alpha \beta \in E_{i}}[\eta _{\alpha }(\eta _{\alpha }+\eta _{\beta })]\log \big [\eta _{\alpha }(\eta _{\alpha }+\eta _{\beta })]. \end{aligned}$$
(11)

  • Fifth Zagreb Entropy

If \(G(\alpha \beta )=\eta _{\beta }(\eta _{\alpha }+\eta _{\beta })\), then

$$\begin{aligned} \sum \limits _{\alpha \beta \in E }G(\alpha \beta )=\sum \limits _{\alpha \beta \in E }\eta _{\beta }(\eta _{\alpha }+\eta _{\beta })=M_{5} \end{aligned}$$

Now Equation (8) represents the fifth Zagreb entropy.

$$\begin{aligned} ENT_{M_{5}}(G)=\log \big (M_{5}\big )-\frac{1}{\big (M_{5}\big )} \sum \limits _{i=1}^{q}\sum \limits _{\alpha \beta \in E_{i}}[\eta _{\beta }(\eta _{\alpha }+\eta _{\beta })]\log \big [\eta _{\beta }(\eta _{\alpha }+\eta _{\beta })]. \end{aligned}$$
(12)

  • Redefined First Zagreb Entropy

If \(G(\alpha \beta )={\frac{\eta _{\alpha }+\eta _{\beta }}{\eta _{\alpha }\times \eta _{\beta }}}\), then

$$\begin{aligned} \sum \limits _{\alpha \beta \in E }G(\alpha \beta )=\sum \limits _{\alpha \beta \in E }{\frac{\eta _{\alpha }+\eta _{\beta }}{\eta _{\alpha }\times \eta _{\beta }}}={ReZG_1} \end{aligned}$$

Now Equation (8) represents the redefined first Zagreb Entropy.

$$\begin{aligned} ENT_{{ReZ_1}}(G)=\log \big (ReZG_1\big )-\frac{1}{\big (ReZG_1\big )} \sum \limits _{i=1}^{q}\sum \limits _{\alpha \beta \in E_{i}}[{\frac{\eta _{\alpha }+\eta _{\beta }}{\eta _{\alpha }\times \eta _{\beta }}}]\log \big [{\frac{\eta _{\alpha }+\eta _{\beta }}{\eta _{\alpha }\times \eta _{\beta }}}\big ]. \end{aligned}$$
(13)

  • Redefined Second Zagreb Entropy

If \(G(\alpha \beta )={\frac{\eta _{\alpha }\times \eta _{\beta }}{\eta _{\alpha }+\eta _{\beta }}}\), then

$$\begin{aligned} \sum \limits _{\alpha \beta \in E }G(\alpha \beta )=\sum \limits _{\alpha \beta \in E }{\frac{\eta _{\alpha }\times \eta _{\beta }}{\eta _{\alpha }+\eta _{\beta }}}=ReZG_{2} \end{aligned}$$

Now Equation (8) represents the redefined second Zagreb entropy.

$$\begin{aligned} ENT_{ReZG_{2}}(G)=\log \big (ReZG_{2}\big )-\frac{1}{\big (ReZG_{2}\big )} \sum \limits _{i=1}^{q}\sum \limits _{\alpha \beta \in E_{i}}[{\frac{\eta _{\alpha }\times \eta _{\beta }}{\eta _{\alpha }+\eta _{\beta }}}]\log \big [{\frac{\eta _{\alpha }\times \eta _{\beta }}{\eta _{\alpha }+\eta _{\beta }}}\big ]. \end{aligned}$$
(14)

  • Redefined third Zagreb Entropy

If \(G(\alpha \beta )={(\eta _{\alpha }\times \eta _{\beta })}{(\eta _{\alpha }+\eta _{\beta })}\), then

$$\begin{aligned} \sum \limits _{\alpha \beta \in E }G(\alpha \beta )=\sum \limits _{\alpha \beta \in E }{(\eta _{\alpha }\times \eta _{\beta })}{(\eta _{\alpha }+\eta _{\beta })}=ReZG_{3} \end{aligned}$$

Now Equation (8) represents the redefined third Zagreb Entropy.

$$\begin{aligned} ENT_{ReZG_{3}}(G)=\log \big (ReZG_{3}\big )-\frac{1}{\big (ReZG_{3}\big )} \sum \limits _{i=1}^{q}\sum \limits _{\alpha \beta \in E_{i}}[{(\eta _{\alpha }\times \eta _{\beta })}{(\eta _{\alpha }+\eta _{\beta })}]\log \big [{(\eta _{\alpha }\times \eta _{\beta })}{(\eta _{\alpha }+\eta _{\beta })}\big ]. \end{aligned}$$
(15)

The majority of silicon dioxide is produced by mining, which also includes quartz purification and sand mining. Quartz may be used for a variety of tasks, but in order to create a purer or more useful material, chemical processing is needed. The main component used in the creation of most glass is silica17. The moment at which freezing occurs Melting silica with other minerals reduces the melting point of the mixture and increases fluidity thanks to the depression idea.19,33. Similar to the odd physical properties of liquid water, molten silica possesses the following properties: A minimal heat capacity, a maximum density at temperatures below \(5000^\circ\)C, and negative temperature expansion34.

The unit structure and copies of the structure of Silicon dioxide are shown in Figure 1. Using Figure 1 we partitioned edges into 8 sets based on the degree of vertices.

$$\begin{aligned} & E_{(2,2)}=\{\alpha \beta \in E(SiO_{2}): \eta _{\alpha }=2, \eta _{\beta }=2\} \\ & \quad E_{(1,2)}=\{\alpha \beta \in E(SiO_{2}): \eta _{\alpha }=1, \eta _{\beta }=2\} \\ & \quad E_{(2,3)}=\{\alpha \beta \in E(SiO_{2}): \eta _{\alpha }=2, \eta _{\beta }=3\} \end{aligned}$$

The cardinality of \(E_{(1,2)}\) is \(4m+4n+4mn\), \(E_{(2,2)}\) is \(-4m-4n+8mn\), and \(E_{(2,3)}\) is 54mn.

Fig. 1
Fig. 1
Full size image

(a) Silicon dioxide \(SiO_{2}\) unit cell; (b) Silicon dioxide for \(m=3=n\).

Main result

We have calculated Zagreb-type indices.

  • First Zagreb Index

Put degree-based partition into Eq. 1, in this way, we derive the first Zagreb index: :

$$\begin{aligned} M_1(G)= & \sum _{i=1}^{3}\sum _{\alpha \beta \in E_{i}(G)}{( \eta _{\alpha }+ \eta _{\beta })}\\= & (1+2)(4m+4n+4mn)+(2+2)(-4m-4n+8mn)+(2+3)(54mn)\\= & (3)(4m+4n+4mn)+(4)(-4m-4n+8mn)+(5)(54mn)\\= & 314mn - 4m - 4n. \end{aligned}$$

  • Second Zagreb Index

Put degree-based partition into Eq. 2, in this way, we derive the second Zagreb index:

$$\begin{aligned} M_2(G)= & \sum _{i=1}^{3}\sum _{\alpha \beta \in E_{i}(G)}{ \eta _{\alpha }\times \eta _{\beta }}\\= & (1\times 2)(4m+4n+4mn)+(2\times 2)(-4m-4n+8mn)+(2\times 3)(54mn)\\= & (2)(4m+4n+4mn)+(4)(-4m-4n+8mn)+(6)(54mn)\\= & 364mn - 8m - 8n. \end{aligned}$$

  • Fourth Zagreb Index

Put degree-based partition into Eq. 3, we obtain the fourth Zagreb index as follows:

$$\begin{aligned} M_4(G)= & \sum _{i=1}^{3}\sum _{\alpha \beta \in E_{i}(G)}\eta _{\alpha }{ (\eta _{\alpha }+ \eta _{\beta })}\\= & 1(1+2)(4m+4n+4mn)+2(2+2)(-4m-4n+8mn)+2(2+3)(54mn)\\= & (3)(4m+4n+4mn)+(8)(-4m-4n+8mn)+(10)(54mn)\\= & 616mn - 20m - 20n. \end{aligned}$$

Put degree-based partition into Eq. 4, we obtain the fifth Zagreb index as follows:

  • Fifth Zagreb Index

$$\begin{aligned} M_5(G)= & \sum _{i=1}^{3}\sum _{\alpha \beta \in E_{i}(G)}\eta _{\beta }{ (\eta _{\alpha }+ \eta _{\beta })}\\= & 2(1+2)(4m+4n+4mn)+2(2+2)(-4m-4n+8mn)+3(2+3)(54mn)\\= & (6)(4m+4n+4mn)+(8)(-4m-4n+8mn)+(15)(54mn)\\= & 898mn - 8m - 8n. \end{aligned}$$

  • First Redefined Zagreb Index

Put degree-based partition into Eq. 5, in this way, we derive the redefined first Zagreb index:

$$\begin{aligned} ReZG_1(G)= & \sum _{i=1}^{3}\sum _{\alpha \beta \in E_{i}(G)}\frac{ \eta _{\alpha }+ \eta _{\beta }}{ \eta _{\alpha }\times \eta _{\beta }}\\= & \left( \frac{1+2}{1\times 2}\right) (4m+4n+4mn)+\left( \frac{2+2}{2\times 2}\right) (-4m-4n+8mn)+\left( \frac{2+3}{2\times 3}\right) (54mn)\\= & \left( \frac{3}{2}\right) (4m+4n+4mn)+\left( \frac{4}{4}\right) (-4m-4n+8mn)+\left( \frac{5}{6}\right) (54mn)\\= & 59mn + 2m + 2n. \end{aligned}$$

  • Second Redefined Zagreb Index

Put degree-based partition into Eq. 6, in this way, we derive the redefined second Zagreb index:

$$\begin{aligned} ReZG_2(G)= & \sum _{i=1}^{3}\sum _{\alpha \beta \in E_{i}(G)}\frac{ \eta _{\alpha }\times \eta _{\beta }}{ \eta _{\alpha }+ \eta _{\beta }}\\= & \left( \frac{1\times 2}{1+2}\right) (4m+4n+4mn)+\left( \frac{2\times 2}{2+2}\right) (-4m-4n+8mn)+\left( \frac{2\times 3}{2+3}\right) (54mn)\\= & \left( \frac{2}{3}\right) (4m+4n+4mn)+\left( \frac{4}{4}\right) (-4m-4n+8mn)+\left( \frac{6}{5}\right) (54mn)\\= & 75.46666667mn - 1.333333333m - 1.333333333n. \end{aligned}$$

  • Third Redefined Zagreb Index

Put degree-based partition into Eq. 7, in this way, we derive the redefined third Zagreb index:

$$\begin{aligned} ReZG_3(G)= & \sum _{i=1}^{3}\sum _{\alpha \beta \in E_{i}(G)}\bigg (( \eta _{\alpha }\times \eta _{\beta })( \eta _{\alpha }+ \eta _{\beta })\bigg )\\ \\= & \left( (1\times 2)(1+2)\right) (4m+4n+4mn)+\left( (2\times 2)(2+2)\right) (-4m-4n+8mn)+\left( (2\times 3)(2+3)\right) (54mn)\\= & (6)(4m+4n+4mn)+(16)(-4m-4n+8mn)+(30)(54mn)\\= & 1772mn - 40m - 40n. \end{aligned}$$
Table 1 Comparison of topological indices numerically.
Full size table
Fig. 2
Fig. 2
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Comparing indices via Radar chart.

A thorough numerical and graphical comparison of different indices when the parameters (m) and (n) are increased is shown in Table 1 and Figure 2. The results unambiguously demonstrate a positive association by showing that all indices rise with greater values of (m) and (n). The indicators’ respective rates of rise, however, differ. The radar chart, in particular, indicates that the \(ReZG_{3}\) index is rising faster than the other indexes. This implies that \(ReZG_{3}\), maybe as a result of its underlying computation technique, is more susceptible to variations in (m) and (n). \(ReZG_{3}\)sharp increase highlights its potential as a highly responsive measure, which may be especially helpful in applications that need quick or high-sensitivity change detection.

  • First Zagreb Entropy

Put down the computed first Zagreb index and edge partition into Eq.9, then the first Zagreb entropy is:

$$\begin{aligned} ENT_{M_{1}}(G)= & \log \big (M_{1}\big )-\frac{1}{\big (M_{1}\big )} \sum \limits _{i=1}^{3}\sum \limits _{\alpha \beta \in E_{i}}[(\eta _{\alpha }+\eta _{\beta })]\log \big [(\eta _{\alpha }+\eta _{\beta })\big ]\\ ENT_{M_{1}}(G)= & \log \big (314mn - 4m - 4n\big )-\frac{(3)(4m+4n+4mn) \log (3)}{\big (314mn - 4m - 4n\big )}-\frac{(4)(-4m-4n+8mn)\log (4)}{\big (314mn - 4m - 4n\big )}\\- & \frac{(5)(54mn)\log (5)}{\big (314mn - 4m - 4n\big )}. \end{aligned}$$

  • Second Zagreb Entropy

Put down the computed second Zagreb index and edge partition into Eq.10, then the second Zagreb entropy is:

$$\begin{aligned} ENT_{M_{2}}(G)= & \log \big (M_{2}\big )-\frac{1}{\big (M_{2}\big )} \sum \limits _{i=1}^{3}\sum \limits _{\alpha \beta \in E_{i}}[(\eta _{\alpha }\times \eta _{\beta })]\log \big [(\eta _{\alpha }\times \eta _{\beta })\big ]\\ \\ ENT_{M_{2}}(G)= & \log \big (364mn - 8m - 8n\big )-\frac{(2)(4m+4n+4mn) \log (2)}{\big (364mn - 8m - 8n\big )}-\frac{(4)(-4m-4n+8mn)\log (4)}{\big (364mn - 8m - 8n\big )}\\- & \frac{(6)(54mn)\log (6)}{\big (364mn - 8m - 8n\big )}. \end{aligned}$$

  • Fourth Zagreb Entropy

Put down the computed fourth Zagreb index and edge partition into Eq.11, then the fourth Zagreb entropy is:

$$\begin{aligned} ENT_{M_{4}}(G)= & \log \big (M_{4}\big )-\frac{1}{\big (M_{4}\big )} \sum \limits _{i=1}^{3}\sum \limits _{\alpha \beta \in E_{i}}[\eta _{\alpha }(\eta _{\alpha }+\eta _{\beta })]\log \big [\eta _{\alpha }(\eta _{\alpha }+\eta _{\beta })]\\ ENT_{M_{4}}(G)= & \log \big (616mn - 20m - 20n\big )-\frac{(3)(4m+4n+4mn) \log (3)}{\big (616mn - 20m - 20n\big )}-\frac{(8)(-4m-4n+8mn)\log (8)}{\big (616mn - 20m - 20n\big )}\\ - & \frac{(10)(54mn)\log (10)}{\big (616mn - 20m - 20n\big )}. \end{aligned}$$

  • Fifth Zagreb Entropy

Put down the computed fifth Zagreb index and edge partition into Eq.12, then the fifth Zagreb entropy is:

$$\begin{aligned} ENT_{M_{5}}(G)= & \log \big (M_{5}\big )-\frac{1}{\big (M_{5}\big )} \sum \limits _{i=1}^{3}\sum \limits _{\alpha \beta \in E_{i}}[\eta _{\beta }(\eta _{\alpha }+\eta _{\beta })]\log \big [\eta _{\beta }(\eta _{\alpha }+\eta _{\beta })]\\ ENT_{M_{5}}(G)= & \log \big (898mn - 8m - 8n\big )-\frac{(6)(4m+4n+4mn) \log (6)}{\big (898mn - 8m - 8n\big )}-\frac{(8)(-4m-4n+8mn)\log (8)}{\big (898mn - 8m - 8n\big )}\\- & \frac{(15)(54mn)\log (15)}{\big (898mn - 8m - 8n\big )} \end{aligned}$$

  • Redefined First Zagreb Entropy

Put down the computed redefined first Zagreb index and edge partition into Eq.13, then the redefined first Zagreb entropy is:

$$\begin{aligned} ENT_{{ReZ_1}}(G)= & \log \big (ReZG_1\big )-\frac{1}{\big (ReZG_1\big )} \sum \limits _{i=1}^{3}\sum \limits _{\alpha \beta \in E_{i}}[{\frac{\eta _{\alpha }+\eta _{\beta }}{\eta _{\alpha }\times \eta _{\beta }}}]\log \big [{\frac{\eta _{\alpha }+\eta _{\beta }}{\eta _{\alpha }\times \eta _{\beta }}}\big ]\\ ENT_{{ReZ_1}}(G)= & \log \big (59mn + 2m + 2n\big )-\frac{(\frac{3}{2})(4m+4n+4mn) \log (\frac{3}{2})}{\big (59mn + 2m + 2n\big )}-\frac{(\frac{4}{4})(-4m-4n+8mn)\log (\frac{4}{4})}{\big (59mn + 2m + 2n\big )}\\- & \frac{(\frac{5}{6})(54mn)\log (\frac{5}{6})}{\big (59mn + 2m + 2n\big )}. \end{aligned}$$

  • Redefined Second Zagreb Entropy

Put down the computed redefined second Zagreb index and edge partition into Eq.14, then the redefined second Zagreb entropy is:

$$\begin{aligned} ENT_{ReZG_{2}}(G)= & \log \big (ReZG_{2}\big )-\frac{1}{\big (ReZG_{2}\big )} \sum \limits _{i=1}^{3}\sum \limits _{\alpha \beta \in E_{i}}[{\frac{\eta _{\alpha }\times \eta _{\beta }}{\eta _{\alpha }+\eta _{\beta }}}]\log \big [{\frac{\eta _{\alpha }\times \eta _{\beta }}{\eta _{\alpha }+\eta _{\beta }}}\big ]\\ ENT_{{ReZ_2}}(G)= & \log \big (75.46666667mn - 1.333333333m - 1.333333333n\big )\\ - & \frac{(\frac{2}{3})(4m+4n+4mn) \log (\frac{3}{2})}{\big (75.46666667mn - 1.333333333m - 1.333333333n\big )}\\- & \frac{(\frac{4}{4})(-4m-4n+8mn)\log (\frac{4}{4})}{\big (75.46666667mn - 1.333333333m - 1.333333333n\big )}\\- & \frac{(\frac{6}{5})(54mn)\log (\frac{6}{5})}{\big (75.46666667mn - 1.333333333m - 1.333333333n\big )}. \end{aligned}$$

  • Redefined Third Zagreb Entropy

Put down the computed redefined third Zagreb index and edge partition into Eq.15, then the redefined third Zagreb entropy is:

$$\begin{aligned} ENT_{ReZG_{3}}(G)= & \log \big (ReZG_{3}\big )-\frac{1}{\big (ReZG_{3}\big )} \sum \limits _{i=1}^{3}\sum \limits _{\alpha \beta \in E_{i}}[{(\eta _{\alpha }\times \eta _{\beta })}{(\eta _{\alpha }+\eta _{\beta })}]\log \big [{(\eta _{\alpha }\times \eta _{\beta })}{(\eta _{\alpha }+\eta _{\beta })}\big ]\\ ENT_{{ReZ_3}}(G)= & \log \big (1772mn - 40m - 40n\big )-\frac{(6)(4m+4n+4mn) \log (6)}{\big (1772mn - 40m - 40n\big )}-\frac{(16)(-4m-4n+8mn)\log (16)}{\big (1772mn - 40m - 40n\big )}\\ - & \frac{(30)(54mn)\log (30)}{\big (1772mn - 40m - 40n\big )}. \end{aligned}$$
Table 2 Comparing entropy measures numerically.
Full size table
Fig. 3
Fig. 3
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Comparing entropy via Radar chart.

A thorough numerical and graphical comparison of different entropy when the parameters (m) and (n) are increased is shown in Table 2 and Figure 3. The results unambiguously demonstrate a positive association by showing that all indices rise with greater values of (m) and (n). The indicators’ respective rates of rise, however, differ. The radar chart, in particular, indicates that the \(ENT_{ReZG3}\) entropy is rising faster than the other entropy. This implies that \(ENT_{ReZG3}\), maybe as a result of its underlying computation technique, is more susceptible to variations in (m) and (n). \(ENT_{ReZG3}\)sharp increase highlights its potential as a highly responsive measure, which may be especially helpful in applications that need quick or high-sensitivity change detection.

Regression model

The logarithmic regression model is a sort of nonlinear regression that is employed in scenarios where the dependent variable has a logarithmic pattern but has little to no association with one or more variables. The dependent variable in this model is the response’s natural logarithm, whereas the independent variable, entropy, is plotted against the independent variable indices. Simply said, the exponential growth or decay phenomena is employed or used in every sector these days, including biology, economics, finance, and many more. The logarithmic regression model has the following form.

$$\begin{aligned} ENT(TI)=a\ln {(TI)}+b. \end{aligned}$$

where ENT(TI) is the dependent variable, (TI) is the independent variable, \(\ln (TI)\) represents the natural logarithm of TI and a and b are coefficients to be estimated from the data. This is particularly helpful when there is a change in the independent variable and a corresponding increase or fall in the dependent variable at different rates during the course of the computed periods. When there are saturation effects or when the rate of returns progressively declines, this is very helpful. When determining associations between variables and creating better prediction models, logarithmic regression may be a better option than line regressions in some situations.

Table 3 Model of logarithmic regression between different indices of entropies.
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Fig. 4
Fig. 4
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Logarithmic regression’s visual behaviour.

Two different logarithmic regression models are summarised in Table 3, and Figure 4. The entropy measure is predicted by these models using the natural logarithm of multiple indices (\(M_{1}\), and \(M_{2}\)). With a \(R^{2}\) value of 1, the model perfectly fits the data. This indicates that the whole fluctuation in the entropy measure can be explained by the natural logarithm of \(M_{1}(G)\). It has a strong linear connection. This indicates that the whole fluctuation in the entropy measure can be explained by the natural logarithm of \(M_{2}(G)\). It has a strong linear connection. Relatively modest, the number (0.433) shows precise coefficient calculations. The strong F statistic (\(5041268.294-6180203.675\)) at a significance level of 0.000 indicates.

Table 4 Model of logarithmic regression between different indices of entropies.
Full size table
Fig. 5
Fig. 5
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Logarithmic regression’s visual behaviour.

Two different logarithmic regression models are summarised in Table 4, and Figure 5. The entropy measure is predicted by these models using the natural logarithm of several indices (\(M_{4}\), and \(M_{5}\)). The \(R^{2}\) squared value of 1 indicates that the model matches the data exactly. This indicates that the whole fluctuation in the entropy measure can be explained by the natural logarithm of \(M_{5}(G)\). It has a strong linear connection. This indicates that the whole fluctuation in the entropy measure can be explained by the natural logarithm of \(M_{2}(G)\). It has a strong linear connection. Relatively modest, the number \((0.432-0.433)\) shows precise coefficient calculations. The strong F statistic (\(1880448.754-53458895.12\)) at a significance level of 0.000.

Table 5 Model of logarithmic regression between different indices of entropies.
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Three different logarithmic regression models are summarised in Table 5, and Figure 6 and Figure 7. The entropy measure is predicted by these models using the natural logarithm of several indices, \(ReZG_{1}\), \(ReZG_{2}\), and \(ReZG_{3}\). The \(R^{2}\) squared value of 1 indicates that the model matches the data exactly. This indicates that the whole fluctuation in the entropy measure can be explained by the natural logarithm of \(ReZG_{1}(G)\) and \(ReZG_{2}(G)\). It has a strong linear connection. This indicates that the whole fluctuation in the entropy measure can be explained by the natural logarithm of \(ReZG_{3}(G)\). It has a strong linear connection. Relatively modest, the number \((0.432-0.433)\) shows precise coefficient calculations. The strong F statistic (\(261751.313-965703871.7\)) at a significance level of 0.000.

Fig. 6
Fig. 6
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Logarithmic regression’s visual behaviour.

Fig. 7
Fig. 7
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\(ReZG_{3}(G)\) VS \(ENT_{ReZG_{3}}(G)\).

Conclusion

Our analysis in this work has focused on the computation of numerous Zagreb-type indices and their link with entropy measurements, as well as the mathematical and computational characterization of silicon dioxide \((SiO_{2})\). The application of topological indices from the chemical graph theory subfield, known as Zagreb indices, can help explain the molecular structure of silicon dioxide. Subsequently, the silicon dioxide entropy value was ascertained, serving as a quantifiable indicator of the unpredictability present in the chemical system. We constructed logarithmic regression models relating entropy measurements to indices. We see that the greatest value of R and R squared is 1, as we can see. It indicates that entropy and indices have a significant relationship.