• **Entropy related to the** *ABC* **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 (1) and Table 1, the atom–bond connectivity index is

$$\begin{aligned} ABC(S(H\_3BO\_3)) \quad &= \sum\_{\tilde{\mathbb{Z}}\_{(1^2-2)}} x^{\sqrt{\frac{1+2-2}{1\times 2}}} + \sum\_{\tilde{\mathbb{Z}}\_{(2^2-2)}} x^{\sqrt{\frac{2+2-2}{2\times 2}}} + \sum\_{\tilde{\mathbb{Z}}\_{(2^2-3)}} x^{\sqrt{\frac{2+3-2}{2\times 3}}} \\ &= -2(s+t+1)x^{\sqrt{\frac{1}{2}}} + 12(st+s+t)x^{\sqrt{\frac{1}{2}}} \\ &+ -6(3s+3t+4st-1)x^{\sqrt{\frac{1}{2}}} \end{aligned} \tag{11}$$

Differentiate (11) at *x* = 1; we get the atom–bond connectivity index

$$ABCS(H\_3BO\_3) = \sqrt{\frac{1}{2}}(32s + 32t + 36st - 4)\tag{12}$$

Here, we determine the atom–bond connectivity entropy by using Table 1 and Equation (12) in Equation (6) according to the following:

**Table 1.** Edge division based on vertices in the layer structure of subdivision *H*3*BO*3(*s*, *t*).


*ENTABCS*(*H*3*BO*3) = log (*ABC*) <sup>−</sup> <sup>1</sup> *ABC* log ∏ *ξ*(1,2) [ (*Vai* + *Vaj* − 2) (*Vai* × *Vaj* ) ] [ (*Vai* <sup>+</sup>*Vaj* −2) (*Vai* <sup>×</sup>*Vaj* ) ] × ∏ *ξ*(2,2) [ (*Vai* + *Vaj* − 2) (*Vai* × *Vaj* ) ] [ (*Vai* <sup>+</sup>*Vaj* −2) (*Vai* <sup>×</sup>*Vaj* ) ] × ∏ *ξ*(2,3) [ (*Vai* + *Vaj* − 2) (*Vai* × *Vaj* ) ] [ (*Vai* <sup>+</sup>*Vaj* −2) (*Vai* <sup>×</sup>*Vaj* ) ] = log ( 1 2 (32*<sup>s</sup>* <sup>+</sup> <sup>32</sup>*<sup>t</sup>* <sup>+</sup> <sup>36</sup>*st* <sup>−</sup> <sup>4</sup>) <sup>−</sup> <sup>1</sup> 1 <sup>2</sup> (32*s* + 32*t* + 36*st* − 4) <sup>×</sup> log <sup>2</sup>(*<sup>s</sup>* <sup>+</sup> *<sup>t</sup>* <sup>+</sup> <sup>1</sup>)(<sup>1</sup> 2 ) 1 <sup>2</sup> <sup>×</sup> <sup>12</sup>(*st* <sup>+</sup> *<sup>s</sup>* <sup>+</sup> *<sup>t</sup>*)(<sup>1</sup> 2 ) 1 2 <sup>×</sup> <sup>6</sup>(3*<sup>s</sup>* <sup>+</sup> <sup>3</sup>*<sup>t</sup>* <sup>+</sup> <sup>4</sup>*st* <sup>−</sup> <sup>1</sup>)(<sup>1</sup> 2 ) 1 2 . (13)
