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

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

$$\begin{aligned} \left(A\_{(\mathbb{G},x)}\left(S(H\_3BO\_3)\right)\right) &= \sum\_{\mathbb{F}\_{(1^\sim 2)}} x^{|1^\sim 2|} + \sum\_{\mathbb{F}\_{(2^\sim 2)}} x^{|2^\sim 2|} + \sum\_{\mathbb{F}\_{(2^\sim 3)}} x^{|2^\sim 3|} \\ &= 2(s+t+1)x + 12(st+s+t) + 6(3s+3t+4st-1)x \end{aligned} \tag{17}$$

Differentiate (17) at *x* = 1; we get the atom–bond connectivity index

$$A\_{(G,x)}S(H\_3BO\_3) = 32s + 32t + 36st - 4 \tag{18}$$

Here, we determine the atom–bond connectivity entropy by using Table 1 and Equation (18) in Equation (9) according to the following:

$$\begin{split} \operatorname{ENT}\_{A(\mathbb{G},t)}S(H\_3BO\_3) &= \log\left(A\_{(\mathbb{G},x)}\right) - \frac{1}{A\_{(\mathbb{G},x)}}\log\left\{\prod\_{\tilde{\xi}\_{(12)}} \left[|V\_{a\_i} - V\_{a\_j}|\right]^{||V\_{a\_i} - V\_{a\_j}||} \right. \\ &\times \prod\_{\tilde{\xi}\_{(22)}} \left[|V\_{a\_i} - V\_{a\_j}|\right]^{||V\_{b\_i} - V\_{a\_j}||} \times \prod\_{\tilde{\xi}\_{(23)}} \left[|V\_{a\_i} - V\_{a\_j}|\right]^{||V\_{b\_i} - V\_{a\_j}||} \right\} \\ &= \log\left(32s + 32t + 36st - 4\right) - \frac{1}{32s + 32t + 36st - 4} \log\left\{2(s + t + 1) \\ &+ \quad 12(st + s + t) + 6(3s + 3t + 4st - 1)\right\}. \end{split} \tag{19}$$
