# On the Inverse Symmetric Division Deg Index of Unicyclic Graphs

^{*}

## Abstract

**:**

## 1. Introduction

**Problem**

**1.**

## 2. Results

**Lemma**

**1**

**.**If G is a graph with m edges and with at least three vertices, then

**Corollary**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Proof.**

**Corollary**

**2.**

**Transformation**

**1.**

- (i).
- ${d}_{v}\ge {d}_{z}$ for every $z\in N\left(v\right)\cup N\left(w\right)$ and
- (ii).
- The vertex w has at least one neighbor (different from v) that is not adjacent to v.

**Lemma**

**4.**

**Proof.**

**Figure 2.**Transformation 1 used in the proof of Lemma 4, when the edge $vw$ does not lie on a cycle.

**Figure 3.**Transformation 1 used in the proof of Lemma 4, when the edge $vw$ lies on the cycle of length greater than 3.

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**1.**

- (i).
- The girth is three.
- (ii).
- Every vertex of the maximum degree lies on the unique cycle.
- (iii).
- All the neighbors of every vertex of the maximum degree are pendent.

**Proof.**

## 3. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Albalahi, A.M.; Ali, A.
On the Inverse Symmetric Division Deg Index of Unicyclic Graphs. *Computation* **2022**, *10*, 181.
https://doi.org/10.3390/computation10100181

**AMA Style**

Albalahi AM, Ali A.
On the Inverse Symmetric Division Deg Index of Unicyclic Graphs. *Computation*. 2022; 10(10):181.
https://doi.org/10.3390/computation10100181

**Chicago/Turabian Style**

Albalahi, Abeer M., and Akbar Ali.
2022. "On the Inverse Symmetric Division Deg Index of Unicyclic Graphs" *Computation* 10, no. 10: 181.
https://doi.org/10.3390/computation10100181