• **Entropy related to the** *ABS* **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 (2) and Table 2, the *ABS* polynomial is

$$\begin{split} ABSL(H\_3BO\_3) &= \sum\_{\tilde{\xi}(2-3)} x^{\sqrt{\frac{2+2}{2+3}}} + \sum\_{\tilde{\xi}(2-4)} x^{\sqrt{\frac{2+4-2}{2+4}}} + \sum\_{\tilde{\xi}(3-3)} x^{\sqrt{\frac{2+2}{3+3}}} + \sum\_{\tilde{\xi}(4-4)} x^{\sqrt{\frac{2+4-2}{3+4}}} \\ &= 6(1+t+s)x^{\sqrt{\frac{2}{3}}} + 2(s+t+1)x^{\sqrt{\frac{2}{3}}} + 4(s+t+3st-2)x^{\sqrt{\frac{2}{3}}} \\ &+ 2(5s+5t+6st-1)x^{\sqrt{\frac{2}{3}}} + 2(s+t+3st-2)x^{\sqrt{\frac{2}{3}}}. \end{split} \tag{26}$$

Taking the first derivative of Equation (26) at *x* = 1, we get the *ABS* index

$$\begin{split} \left( \mathrm{ABS} (\mathrm{L} (\mathrm{H\_3BO\_3}) \right) &= & 6(1+t+s) \sqrt{\frac{3}{5}} + 2(\mathrm{6st}+3s+3t-3) \sqrt{\frac{2}{3}} + 2(\mathrm{5s}+5t+6st-1) \sqrt{\frac{5}{7}} \\ &+ & 2(s+t+3st-2) \sqrt{\frac{3}{4}}. \end{split} \tag{27}$$

Here, we determine the *ABS* entropy by using Table 2 and Equation (27) in Equation (6) according to the following:

*ENTABS*(*L*(*H*3*BO*3)) = log (*ABS*) <sup>−</sup> <sup>1</sup> *ABS* log ∏ *ξ*(2,3) [ (*Vai* + *Vaj* − 2) (*Vai* + *Vaj* ) ] [ (*Vai* <sup>+</sup>*Vaj* −2) (*Vai* <sup>+</sup>*Vaj* ) ] × ∏ *ξ*(2,4) [ (*Vai* + *Vaj* − 2) (*Vai* + *Vaj* ) ] [ (*Vai* <sup>+</sup>*Vaj* −2) (*Vai* <sup>+</sup>*Vaj* ) ] × ∏ *ξ*(3,3) [ (*Vai* + *Vaj* − 2) (*Vai* + *Vaj* ) ] [ (*Vai* <sup>+</sup>*Vaj* −2) (*Vai* <sup>+</sup>*Vaj* ) ] × ∏ *ξ*(3,4) [ (*Vai* + *Vaj* − 2) (*Vai* + *Vaj* ) ] [ (*Vai* <sup>+</sup>*Vaj* −2) (*Vai* <sup>+</sup>*Vaj* ) ] × ∏ *ξ*(4,4) [ (*Vai* + *Vaj* − 2) (*Vai* + *Vaj* ) ] [ (*Vai* <sup>+</sup>*Vaj* −2) (*Vai* <sup>+</sup>*Vaj* ) ] <sup>=</sup> log (*ABS*) <sup>−</sup> <sup>1</sup> *ABS* log <sup>6</sup>(<sup>1</sup> <sup>+</sup> *<sup>t</sup>* <sup>+</sup> *<sup>s</sup>*)(<sup>3</sup> 5 ) 3 <sup>5</sup> <sup>+</sup> <sup>2</sup>(*<sup>s</sup>* <sup>+</sup> *<sup>t</sup>* <sup>+</sup> <sup>1</sup>)(<sup>2</sup> 3 ) 2 3 <sup>+</sup> <sup>4</sup>(*<sup>s</sup>* <sup>+</sup> *<sup>t</sup>* <sup>+</sup> <sup>3</sup>*st* <sup>−</sup> <sup>2</sup>)(<sup>2</sup> 3 ) 2 <sup>3</sup> <sup>+</sup> <sup>2</sup>(5*<sup>s</sup>* <sup>+</sup> <sup>5</sup>*<sup>t</sup>* <sup>+</sup> <sup>6</sup>*st* <sup>−</sup> <sup>1</sup>)(<sup>5</sup> 7 ) 5 7 <sup>+</sup> <sup>2</sup>(*<sup>s</sup>* <sup>+</sup> *<sup>t</sup>* <sup>+</sup> <sup>3</sup>*st* <sup>−</sup> <sup>2</sup>)(<sup>3</sup> 4 ) 3 4 . (28)
