• **Entropy related to the** *ABC* **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 (1) and Table 2, the *ABC* polynomial is

$$\begin{split} \text{ABCL(H\_3BO\_3)} &= \sum\_{\tilde{\xi}\_{(2\sim 3)}} x^{\sqrt{\frac{2+4-2}{2+3}}} + \sum\_{\tilde{\xi}\_{(2\sim 4)}} x^{\sqrt{\frac{2+4-2}{2+3}}} + \sum\_{\tilde{\xi}\_{(3\sim 3)}} x^{\sqrt{\frac{2+3-2}{3+4}}} + \sum\_{\tilde{\xi}\_{(3\sim 4)}} x^{\sqrt{\frac{3+4-2}{3+4}}} \\ &= \ 6(1+t+s)x^{\sqrt{\frac{1}{2}}} + 2(s+t+1)x^{\frac{1}{2}} + 4(s+t+3st-2)x^{\frac{2}{3}} \\ &+ \ 2(5s+5t+6st-1)x^{\sqrt{\frac{5}{12}}} + 2(s+t+3st-2)x^{\sqrt{\frac{3}{8}}}. \end{split} \tag{23}$$

Taking the first derivative of Equation (23) at *x* = 1, we get the *ABC* index

$$\begin{aligned} \text{ABCD}(H\_3BO\_3) &= & 6(1+t+s)\sqrt{\frac{1}{2}} + \frac{2}{3}(24st+11s+11t-5) + 2(5s+5t+6st-1)\sqrt{\frac{5}{12}} \\ &+ & 2(s+t+3st-2)\sqrt{\frac{3}{8}} \end{aligned} \tag{24}$$

Here, we determine the *ABC* entropy by using Table 2 and Equation (24) in Equation (6) according to the following:

*ENT* <sup>=</sup> log (*ABC*) <sup>−</sup> <sup>1</sup> *ABC* 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 (*ABC*) <sup>−</sup> <sup>1</sup> *ABS* log <sup>6</sup>(<sup>1</sup> <sup>+</sup> *<sup>t</sup>* <sup>+</sup> *<sup>s</sup>*)(<sup>1</sup> 2 ) √1 <sup>2</sup> <sup>+</sup> <sup>2</sup>(*<sup>s</sup>* <sup>+</sup> *<sup>t</sup>* <sup>+</sup> <sup>1</sup>)( <sup>1</sup> 2 ) 1 2 <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> 12 ) 5 12 <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> 8 ) 3 8 . (25)


#### **Table 2.** Edge division based on vertices in the line graph *H*3*BO*3(*s*, *t*) layer structure.
