# Remarks on Distance Based Topological Indices for ℓ-Apex Trees

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## Abstract

**:**

## 1. Introduction

## 2. One Extremum

**Theorem**

**1.**

- G has the minimum possible value of I if I is decreasing;
- G has the maximum possible value of I if I is increasing.

**Proof.**

#### 2.1. Generalized Wiener Index

**Corollary**

**1.**

- If $\lambda >0$ then ${W}^{\lambda}\left(G\right)$ has the minimum value if and only if $G={K}_{\ell}+T$, where T is any tree on $n-\ell $ vertices;
- If $\lambda <0$ then ${W}^{\lambda}\left(G\right)$ has the maximum value if and only if $G={K}_{\ell}+T$, where T is any tree on $n-\ell $ vertices.

**Proof.**

#### 2.1.1. Wiener Index

#### 2.1.2. Harary Index

#### 2.1.3. Hyper-Wiener Index

#### 2.2. Connective Eccentricity Index

**Corollary**

**2.**

**Proof.**

#### 2.3. Generalized Degree Distance Index

**Corollary**

**3.**

**Proof.**

#### Additively Weighted Harary Index

## 3. Other Extremum

**Theorem**

**2.**

- G has the maximum possible value of I if I is decreasing;
- G has the minimum possible value of I if I is increasing.

**Proof.**

#### Generalized Wiener Index

**Corollary**

**4.**

- If $\lambda >0$ then ${W}^{\lambda}\left(G\right)$ has the maximum value if and only if $G={D}_{n-4}(3,1)$;
- If $\lambda <0$ then ${W}^{\lambda}\left(G\right)$ has the minimum value if and only if $G={D}_{n-4}(3,1)$.

**Proof.**

## 4. Further Work

- Modified generalized degree distance. This index is defined as$${H}_{*\lambda}\left(G\right)=\sum _{u\ne v}deg\left(u\right)deg\left(v\right){\mathrm{dist}}^{\lambda}(u,v)$$
- Maximum of Wiener index for bigger ℓ. Let G be an ℓ-apex tree on n vertices, where $\ell \ge 3$ and $n\ge \ell +1$, such that G has maximum Wiener index. It seems that G is the balanced dumbbell graph. i.e. $G\cong {D}_{c}(a,b)$, where $a=\lceil \ell /2\rceil $, $b=\lfloor \ell /2\rfloor $, and $c=n-\ell $.
- Minimum of connective eccentricity index for bigger ℓ. For $n\ge \ell +4$, let $C(n,\ell )$ be the graph obtained from a path ${v}_{1}{v}_{2}\cdots {v}_{n-\ell}$ by connecting ${v}_{2}$ and ${v}_{3}$ to every vertex of a stable set of size $n-\ell $. See Figure 3. It is easy to see that $C(n,\ell )$ is an ℓ-apex graph under assumption that $n-\ell \ge 4$. It seems graphs $C(n,\ell )$ and ${D}_{n-\ell -1}(\ell +2,1)$ are good candidates for the smallest possible value for the connective eccentricity index.

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Knor, M.; Imran, M.; Jamil, M.K.; Škrekovski, R.
Remarks on Distance Based Topological Indices for *ℓ*-Apex Trees. *Symmetry* **2020**, *12*, 802.
https://doi.org/10.3390/sym12050802

**AMA Style**

Knor M, Imran M, Jamil MK, Škrekovski R.
Remarks on Distance Based Topological Indices for *ℓ*-Apex Trees. *Symmetry*. 2020; 12(5):802.
https://doi.org/10.3390/sym12050802

**Chicago/Turabian Style**

Knor, Martin, Muhammad Imran, Muhammad Kamran Jamil, and Riste Škrekovski.
2020. "Remarks on Distance Based Topological Indices for *ℓ*-Apex Trees" *Symmetry* 12, no. 5: 802.
https://doi.org/10.3390/sym12050802