# The Difference of Zagreb Indices of Halin Graphs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**(OP1)**and

**(OP2)**, as shown in Figure 1:

**(OP1)**- Contracting the edge ${v}_{0}{u}_{0}$ into a vertex ${v}_{0}^{\prime}$, and adding a leaf ${u}_{0}^{\prime}$ at ${v}_{0}^{\prime}$ to form ${T}^{\prime}$;
**(OP2)**- Set ${C}^{\prime}={v}_{1}{v}_{2}\cdots {v}_{k-1}{u}_{0}^{\prime}{v}_{k}\cdots {v}_{t}{v}_{1}$.

**Proof.**

**Lemma**

**4.**

**Lemma**

**5.**

## 3. General Halin Graphs

**Theorem**

**1.**

**Proof.**

**Lemma**

**6.**

- (1)
- If n is even, then $m\ge \frac{3}{2}n$, where the equality attains if and only if G is 3-regular.
- (2)
- If n is odd, then $m\ge \frac{1}{2}(3n+1)$, where the equality attains if and only if G is special.

**Proof.**

**Lemma**

**7.**

- (1)
- If n is even, then there is a 3-regular Halin graph on n vertices.
- (2)
- If n is odd, then there is a special Halin graph on n vertices.

**Proof.**

**Lemma**

**8.**

**Lemma**

**9.**

**Proof.**

**Theorem**

**2.**

- (1)
- If n is even, then $\Delta \mathrm{M}\left(G\right)\ge \frac{9n}{2}$, where the equality holds if and only if G is 3-regular.
- (2)
- If n is odd, then $\Delta \mathrm{M}\left(G\right)\ge \frac{9n}{2}+\frac{19}{2}$, where the equality holds if and only if G is special.

**Proof.**

## 4. Halin Graphs with Fewer Inner Vertices

#### 4.1. Halin Graphs with Two Inner Vertices

**Theorem**

**3.**

- (1)
- $\Delta \mathrm{M}\left(G\right)\le 2{n}^{2}-12n+27$, where the equality holds for $p=3$ and $q=n-3$.
- (2)
- If n is even, then $\Delta \mathrm{M}\left(G\right)\ge \frac{5}{4}{n}^{2}-3n$, where the equality holds for $p=q=\frac{n}{2}$; If n is odd, then $\Delta \mathrm{M}\left(G\right)\ge \frac{5}{4}{n}^{2}-3n+\frac{3}{4}$, where the equality holds for $p=\frac{n-1}{2}$ and $q=\frac{n+1}{2}$.

**Proof.**

#### 4.2. Halin Graphs with Three Inner Vertices

**Lemma**

**10.**

**Proof.**

**Lemma**

**11.**

**Proof.**

**Theorem**

**4.**

- (1)
- If $q\ge p$, then $p=3$ and $q=n-5;$
- (2)
- If $p>q$, then $q=3$ and $p=n-5$.

**Proof.**

**Lemma**

**12.**

**Proof.**

**Theorem**

**5.**

- (1)
- If $n\equiv 0\phantom{\rule{4pt}{0ex}}\left(\mathrm{mod}8\right)$, then $\u03f5=1;$
- (2)
- If $n\equiv 1,7\phantom{\rule{4pt}{0ex}}\left(\mathrm{mod}8\right)$, then $\u03f5=-\frac{1}{8};$
- (3)
- If $n\equiv 2,6\phantom{\rule{4pt}{0ex}}\left(\mathrm{mod}8\right)$, then $\u03f5=\frac{1}{2};$
- (4)
- If $n\equiv 3,5\phantom{\rule{4pt}{0ex}}\left(\mathrm{mod}8\right)$, then $\u03f5=\frac{7}{8};$
- (5)
- If $n\equiv 4\phantom{\rule{4pt}{0ex}}\left(\mathrm{mod}8\right)$, then $\u03f5=0$.

**Proof.**

## 5. Conclusions

**Problem**

**1.**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Graphs attaining the maximum value of the difference of Zagreb indices in ${\mathcal{H}}_{12}^{3}$.

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**MDPI and ACS Style**

Zheng, L.; Wang, Y.; Wang, W.
The Difference of Zagreb Indices of Halin Graphs. *Axioms* **2023**, *12*, 450.
https://doi.org/10.3390/axioms12050450

**AMA Style**

Zheng L, Wang Y, Wang W.
The Difference of Zagreb Indices of Halin Graphs. *Axioms*. 2023; 12(5):450.
https://doi.org/10.3390/axioms12050450

**Chicago/Turabian Style**

Zheng, Lina, Yiqiao Wang, and Weifan Wang.
2023. "The Difference of Zagreb Indices of Halin Graphs" *Axioms* 12, no. 5: 450.
https://doi.org/10.3390/axioms12050450