• **Entropy related to the Albertson index of** *L***(***H***3***BO***3)**

Let *L*(*H*3*BO*3) be a line graph of *H*3*BO*3(*s*, *t*)). Then by using Equation (3) and Table 2, the Albertson index is

$$\begin{split} A\_{\text{(G,x)}}L(H\_3BO\_3) &= \sum\_{\tilde{\mathbb{F}}\_{(2-3)}} \mathbf{x}^{|2-3|} + \sum\_{\tilde{\mathbb{F}}\_{(2-4)}} \mathbf{x}^{|2-4|} + \sum\_{\tilde{\mathbb{F}}\_{(3-3)}} \mathbf{x}^{|3-3|} + \sum\_{\tilde{\mathbb{F}}\_{(3-4)}} \mathbf{x}^{|3-4|} + \sum\_{\tilde{\mathbb{F}}\_{(4-4)}} \mathbf{x}^{|4-4|} \\ &= -6(1+t+s)\mathbf{x} + 2(s+t+1)\mathbf{x}^2 + 4(s+t+3st-2) \\ &+ \ 2(5s+5t+6st-1)\mathbf{x} + 2(s+t+3st-2). \end{split} \tag{29}$$

Taking the first derivative of Equation (29) at *x* = 1, we get the Albertson index

$$(A\_{(G,x)}(L(H\text{3BO3})) \quad = \ 2(15st + 13s + 13t - 2) \tag{30}$$

Here, we determine the *A* entropy by using Table 2 and Equation (30) in Equation (9) according to the following:

$$\begin{split}ENT\_{A\_{\{G\_{5}\}}}(L(H\_{5}BO\_{5})) &= \ \log\left(A\right) - \frac{1}{A}\log\left\{\prod\big[\big[V\_{d\_{i}} - V\_{q\_{j}}\big]\big]^{\left[V\_{d\_{i}} - V\_{s\_{j}}\right]}} \\ &\times \prod\limits\_{\tilde{\xi}\in(2,4)} \big[\big[V\_{d\_{i}} - V\_{q\_{j}}\big]\big]^{\left[V\_{d\_{i}} - V\_{q\_{j}}\right]} \times \prod\limits\_{\tilde{\xi}\in(3,4)} \big[\big[V\_{d\_{i}} - V\_{q\_{j}}\big]\big]^{\left[V\_{d\_{i}} - V\_{s\_{j}}\right]} \\ &\times \prod\limits\_{\tilde{\xi}\in(4,4)} \big[\big[V\_{d\_{i}} - V\_{q\_{j}}\big]\big]^{\left[V\_{d\_{i}} - V\_{q\_{j}}\right]} \times \prod\limits\_{\tilde{\xi}\in(4,4)} \big[\big[V\_{d\_{i}} - V\_{q\_{j}}\big]\big]^{\left[V\_{d\_{i}} - V\_{s\_{j}}\right]} \Bigg] \\ &= \ \log 2(15st + 13s + 13t - 2) - \frac{1}{2(15st + 13s + 13t - 2)} \log\left\{6(1 + t + s) \\ &+ \ 4(s + t + 1) + 4(s + t + 3st - 2) + 2(5s + 5t + 6st - 1) \\ &+ \ 2(s + t + 3st - 2)\right\}. \tag{31} \end{split} \tag{32}$$

### • **Entropy related to the** *IRM* **index of** *L*(*H*3*BO*3)

Let *L*(*H*3*BO*3) be a line graph of *H*3*BO*3(*s*, *t*). Then by using Equation (4) and Table 2, the *IRM* index is

$$\begin{split} IRM\_{(G,x)}L(H\_3BO\_3) &= \sum\_{\xi\_{(2\succ 3)}} x^{[2-3]^2} + \sum\_{\xi\_{(2\succ 4)}} x^{[2-4]^2} + \sum\_{\xi\_{(3\succ 3)}} x^{[3-3]^2} + \sum\_{\xi\_{(3\succ 4)}} x^{[3-4]^2} + \sum\_{\xi\_{(4\succ 4)}} x^{[4-4]^2} \\ &= \ 6(1+t+s)x + 2(s+t+1)x^4 + 4(s+t+3st-2) \\ &\quad + \ 2(5s+5t+6st-1)x + 2(s+t+3st-2). \end{split} \tag{32}$$

Taking the first derivative of Equation (32) at *x* = 1, we get the *IRM* index

$$IRM\_{\left(G,x\right)}\left(L\left(H\_3BO\_3\right)\right) \quad = \quad \mathfrak{A}0\text{s} + \mathfrak{A}0t + \mathfrak{A}0st.\tag{33}$$

Here, we determine the *IRM* entropy by using Table 2 and Equation (33) in Equation (10) according to the following:

$$\begin{split}{LENT}\_{IRM}(L(H\_{3}BO\_{3})) &= \ & \log\left(IRM\right) - \frac{1}{IRM}\log\left\{\prod\limits\_{\xi\_{(2,3)}} [\lvert V\_{\xi\_{i}} - V\_{\theta\_{j}}\rvert^{2}]^{\lfloor\lvert V\_{\xi\_{i}} - V\_{\theta\_{j}}\rvert^{2}\rfloor}\right. \\ & \times \ & \left. \begin{aligned} & \times \prod\limits\_{\xi\_{(2,4)}} [\lvert V\_{a\_{i}} - V\_{\theta\_{i}}\rvert^{2}]^{\lfloor\lvert V\_{\xi\_{i}} - V\_{\theta\_{j}}\rvert^{2}\rfloor} \times & \prod\limits\_{\xi\_{(3,3)}} [\lvert V\_{a\_{i}} - V\_{a\_{j}}\rvert^{2}]^{\lfloor\lvert V\_{\xi\_{i}} - V\_{\theta\_{j}}\rvert^{2}\rfloor} \\ & \times & \prod\limits\_{\xi\_{(3,4)}} [\lvert V\_{a\_{i}} - V\_{\theta\_{i}}\rvert^{2}]^{\lfloor\lvert V\_{\xi\_{i}} - V\_{\theta\_{j}}\rvert^{2}\rfloor} \times \prod\limits\_{\xi\_{(4,4)}} [\lvert V\_{a\_{i}} - V\_{a\_{j}}\rvert]^{\lfloor\lvert V\_{\xi\_{i}} - V\_{\theta\_{j}}\rvert\rfloor} \end{aligned} \right. \\ &= & \log\left(30s + 30t + 30st\right) - \frac{1}{30s + 30t + 30st} \log\left\{\delta(1 + t + s) + 8(s + t + 1) \\ & + \ 4(s + t + 3st - 2) + 2(5s + 5t + 6st - 1) + 2(s + t + 3st - 2)\right\}. \end{split}$$

#### **3. Comparison and Conclusions**

Here, molecular descriptors for the subdivision and line graph of the layer structure of *H*3*BO*<sup>3</sup> that are multiplicative and degree-based have been studied. Using these molecular descriptors, we compute the ABC entropy, ABS entropy, A entropy, and IRM entropy of the subdivision and line graph of the layer structure of *H*3*BO*3. Our results (entropies) help to describe the randomness and disorder of a molecule of *H*3*BO*<sup>3</sup> based on the number of different arrangements available to it in a given system or reaction. For instance, the atom–bond connectivity (ABC) index offers excellent calculations of the strain energy of molecules via correlation. When the temperatures of the production of alkanes are described using the ABC-index, a good quantitative structure–property relationship (QSPR) model (r = 0.9970) is produced.

The values of four degree-based indices, namely, the *ABC*-index, *ABS*-index, *A*-index, and *IRM*-index, are presented in this work, numerically in Table 3 and graphically in Figure 5. As shown in Figure 5, the values of all indices are directly proportional to the values of (*s*, *t*), with the values of (*s*, *t*) along the x-axes and the resultant of the indices along the y-axes. The disparities between each topological index for a certain structure are revealed by these charts. The results of the computations demonstrate that the degree-based indices and entropy estimates depend greatly on the values of *s* and *t* or the molecular structure.


**Table 3.** Numerical comparison of molecular descriptors.

**Figure 5.** Graphical Comparison of *ABC*-index, *ABS*-index, Albertson index and *IRM*-index.

**Author Contributions:** Conceptualization, M.U.G.; Software, A.E.O.; Investigation, M.U.G.; Resources, M.C.; Writing—original draft, M.U.G.; Project administration, M.K.M.; Funding acquisition, R.G. These authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** No data were used to support this study.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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