# Extremal Trees with Respect to the Difference between Atom-Bond Connectivity Index and Randić Index

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary Results

**Theorem**

**1**

- (1)
- Suppose that $n\equiv 0\phantom{\rule{0.166667em}{0ex}}\left(mod\phantom{\rule{0.166667em}{0ex}}3\right)$.
- (1.1)
- For $n\ge 9$, the maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{3+2\sqrt{2}-3\sqrt{6}}{4},$$
- (1.2)
- For $n\ge 21$, the second maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{4\sqrt{15}-7\sqrt{6}-4\sqrt{3}+7}{4},$$
- (1.3)
- For $n\ge 21$, the third maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{4\sqrt{15}-4\sqrt{3}-9\sqrt{6}+11}{6},$$

- (2)
- Suppose that $n\equiv 1\phantom{\rule{0.166667em}{0ex}}\left(mod\phantom{\rule{0.166667em}{0ex}}3\right)$.
- (2.1)
- For $n\ge 13$, the maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{11+6\sqrt{15}-4\sqrt{3}-13\sqrt{6}}{12},$$
- (2.2)
- For $n\ge 13$, the second maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{12\sqrt{2}-13\sqrt{6}-4\sqrt{3}+17}{12},$$
- (2.3)
- For $n\ge 25$, the third maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{12\sqrt{15}-6\sqrt{2}-25\sqrt{6}-16\sqrt{3}+29}{12},$$

- (3)
- Suppose that $n\equiv 2\phantom{\rule{0.166667em}{0ex}}\left(mod\phantom{\rule{0.166667em}{0ex}}3\right)$.
- (3.1)
- For $n\ge 5$, the maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{4\sqrt{3}-5\sqrt{6}+1}{12},$$
- (3.2)
- For $n\ge 17$, the second maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{6\sqrt{15}+6\sqrt{2}-17\sqrt{6}-8\sqrt{3}+19}{12},$$
- (3.3)
- For $n\ge 29$, the third maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{18\sqrt{15}-29\sqrt{6}-20\sqrt{3}+31}{12},$$

## 3. Maximum $\mathit{ABC}-\mathit{R}$ Index for Chemical Trees

**Theorem**

**2.**

- (1)
- Suppose that $n\equiv 0\phantom{\rule{0.166667em}{0ex}}\left(mod\phantom{\rule{0.166667em}{0ex}}3\right)$.
- (1.1)
- For $n\ge 21$, the fourth maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{18\sqrt{15}+36\sqrt{2}-36\sqrt{3}-63\sqrt{6}+81}{36},$$
- (1.2)
- For $n\ge 33$, the fifth maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{54\sqrt{15}+18\sqrt{2}-72\sqrt{3}-99\sqrt{6}+117}{36},$$
- (1.3)
- For $n\ge 33$, the sixth maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{24\sqrt{2}-12\sqrt{3}-21\sqrt{6}+33}{12},$$

- (2)
- Suppose that $n\equiv 1\phantom{\rule{0.166667em}{0ex}}\left(mod\phantom{\rule{0.166667em}{0ex}}3\right)$.
- (2.1)
- For $n\ge 37$, the fourth maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{72\sqrt{15}-84\sqrt{3}-111\sqrt{6}+123}{36},$$
- (2.2)
- For $n\ge 37$, the fifth maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{24\sqrt{15}+18\sqrt{2}-36\sqrt{3}-66\sqrt{6}+90}{36},$$
- (2.3)
- For $n\ge 37$, the sixth maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{18\sqrt{15}+54\sqrt{2}-48\sqrt{3}-75\sqrt{6}+105}{36},$$

- (3)
- Suppose that $n\equiv 2\phantom{\rule{0.166667em}{0ex}}\left(mod\phantom{\rule{0.166667em}{0ex}}3\right)$.
- (3.1)
- For $n\ge 29$, the fourth maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{18\sqrt{2}-8\sqrt{3}-17\sqrt{6}+25}{12},$$
- (3.2)
- For $n\ge 29$, the fifth maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{42\sqrt{15}-78\sqrt{6}-48\sqrt{3}+96}{36},$$
- (3.3)
- For $n\ge 29$, the sixth maximum $ABC-R$ value is$$\frac{4\sqrt{3}+\sqrt{6}-5}{12}n+\frac{36\sqrt{15}+36\sqrt{2}-60\sqrt{3}-87\sqrt{6}+111}{36},$$

**Proof.**

- when ${x}_{1,2}\ge 1$,$$\theta \ge \frac{8\sqrt{3}-\sqrt{6}-7}{12}\approx 0.367243>0.080294,$$
- when ${x}_{1,3}\ge 1$,$$\theta \ge \frac{32\sqrt{3}-13\sqrt{6}-19}{36}\approx 0.127285>0.080294,$$
- when ${x}_{2,2}\ge 1$,$$\theta \ge \frac{4\sqrt{3}+\sqrt{6}-6\sqrt{2}+1}{12}\approx 0.157701>0.080294,$$
- when ${x}_{2,3}\ge 2$,$$\theta \ge 2\xb7\frac{8\sqrt{3}+11\sqrt{6}-18\sqrt{2}-13}{36}\approx 0.130275>0.080294.$$

**Case 1.**${x}_{2,3}=1$.

**Case 2.**${x}_{2,3}=0$.

**Subcase 2.1.**${x}_{3,3}=1$.

**Subcase 2.2.**${x}_{3,3}=0$.

- (i)
- If $n\equiv 0\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.166667em}{0ex}}3\right)$, then the fourth, fifth and sixth minimum $\theta $ values are 0.0580155, 0.0736795 and 0.0802936, respectively.
- (ii)
- If $n\equiv 1\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.166667em}{0ex}}3\right)$, then the fourth, fifth and sixth minimum $\theta $ values are 0.0714748, 0.0737492 and 0.0780889, respectively.
- (iii)
- If $n\equiv 2\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.166667em}{0ex}}3\right)$, then the fourth, fifth and sixth minimum $\theta $ values are 0.0602202, 0.0715445 and 0.07588419, respectively.

## 4. Upper Bound for $\mathit{ABC}-\mathit{R}$ Index of Molecular Trees

**Theorem**

**3**

**.**Let T be a molecular tree with n vertices, ${n}_{1}\ge 5$ of which are pendant vertices. Then

**Theorem**

**4.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Chemical trees with the fourth (A), the fifth (B) and the sixth (C) maximum $ABC-R$ values in Theorem 2-(1).

**Figure 2.**Chemical trees with the fourth (D), the fifth (E) and the sixth (F) maximum $ABC-R$ values in Theorem 2-(2).

**Figure 3.**Chemical trees with the fourth (G), the fifth (H) and the sixth (I) maximum $ABC-R$ values in Theorem 2-(3).

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**MDPI and ACS Style**

Zuki, W.N.N.N.W.; Du, Z.; Kamran Jamil, M.; Hasni, R.
Extremal Trees with Respect to the Difference between Atom-Bond Connectivity Index and Randić Index. *Symmetry* **2020**, *12*, 1591.
https://doi.org/10.3390/sym12101591

**AMA Style**

Zuki WNNNW, Du Z, Kamran Jamil M, Hasni R.
Extremal Trees with Respect to the Difference between Atom-Bond Connectivity Index and Randić Index. *Symmetry*. 2020; 12(10):1591.
https://doi.org/10.3390/sym12101591

**Chicago/Turabian Style**

Zuki, Wan Nor Nabila Nadia Wan, Zhibin Du, Muhammad Kamran Jamil, and Roslan Hasni.
2020. "Extremal Trees with Respect to the Difference between Atom-Bond Connectivity Index and Randić Index" *Symmetry* 12, no. 10: 1591.
https://doi.org/10.3390/sym12101591