• **Entropy related to the** *IRM* **index of subdivision** *H***3***BO***<sup>3</sup>**

Let *S*(*H*3*BO*3) be a subdivision of *H*3*BO*3(*s*, *t*). Then by using Equation (4) and Table 1, the atom–bond connectivity index is

$$\begin{aligned} \operatorname{IRM}\_{\left(\mathbf{G},\mathbf{x}\right)}\left(S(H\_3BO\_3)\right) &= \sum\_{\tilde{\mathbb{F}}\_{\left(1\sim2\right)}} \mathbf{x}^{\left[1-2\right]^2} + \sum\_{\tilde{\mathbb{F}}\_{\left(2\sim2\right)}} \mathbf{x}^{\left[2-2\right]^2} + \sum\_{\tilde{\mathbb{F}}\_{\left(2\sim3\right)}} \mathbf{x}^{\left[2-3\right]^2} \\ &= \ 2(s+t+1)\mathbf{x} + 12(st+s+t) \\ &+ \ 6(3s+3t+4st-1)\mathbf{x} \end{aligned} \tag{20}$$

Differentiate (20) at *x* = 1; we get the atom–bond connectivity index

$$IRM\_{(G,x)} \backslash S(H\_3BO\_3) = 32s + 32t + 36st - 4 \tag{21}$$

Here, we determine the atom–bond connectivity entropy by using Table 1 and Equation (21) in Equation (10) according to the following:

$$\begin{split} \operatorname{ENT}\_{IRM\_{\left(\xi,x\right)}}S(H\_{3}BO\_{3}) &= \ \log\left(IRM\_{\left(\xi,x\right)}\right) - \frac{1}{IRM\_{\left(\xi,x\right)}}\log\left\{\prod\_{\xi} \left[\left|V\_{a\_{i}} - V\_{a\_{j}}\right|^{2}\right]^{\left[\left|V\_{a\_{i}} - V\_{a\_{j}}\right|^{2}\right]} \right. \\ &\times \prod\_{\xi\_{\left(2,2\right)}} \left[\left|V\_{a\_{i}} - V\_{a\_{j}}\right|^{2}\right]^{\left[\left|V\_{b\_{i}} - V\_{a\_{j}}\right|^{2}\right]} \times \prod\_{\xi\_{\left(2,3\right)}} \left[\left|V\_{a\_{i}} - V\_{a\_{j}}\right|^{2}\right]^{\left[\left|V\_{b\_{i}} - V\_{a\_{j}}\right|^{2}\right]} \right. \\ &= \log\left(32s + 32t + 36st - 4\right) - \frac{1}{32s + 32t + 36st - 4} \log\left\{2\left(s + t + 1\right) \\ &+ \quad 12\left(st + s + t\right) + 6\left(3s + 3t + 4st - 1\right) \right\}. \end{split} \tag{22}$$

#### *2.2. Layer Structure of H*3*BO*<sup>3</sup> *in the Form of a Line Graph*

In the line graph of the layer structure *H*3*BO*3(*s*, *t*), the atom–bond *E*(*G*) is divided into five groups based on the degree of each edge's end vertices. The set that is disjointed is shown by the symbols *ξ*(*d*(*ui*),*d*(*Vj*)). The first set that is disjointed is *ξ*(2,3), the second set that is disjoint is *ξ*(2,4), the third set that is disjointed is *ξ*(3,3), the fourth set that is disjointed is *ξ*(3,4), and the fifth set that is disjointed is *ξ*(4,4).

Figure 4 displays the *H*3*BO*3(*s*, *t*) layer structure as a line graph.

**Figure 4.** Line graph of *H*3*BO*3.
