# Third Smallest Wiener Polarity Index of Unicyclic Graphs

^{1}Anhui Province Key Laboratory of Animal Nutritional Regulation and Health, Anhui Science and Technology University, Fengyang, China^{2}School of Mathematical Sciences, Anhui University, Hefei, China^{3}Mathematics and Applied Mathematics, Reading Academy Nanjing University of Information Science and Technology, Nanjing, China^{4}Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, China^{5}Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, China

The Wiener polarity index *G* is the number of unordered pairs of vertices *u* and *v* is 3. In this paper, we determine the third smallest Wiener polarity index of unicyclic graphs. Moreover, the corresponding extremal graphs are characterized.

## 1. Introduction

Graph theory is one of the most special and unique branches of mathematics. Recently, it has attained much attention among researchers because of its wide range of applications in computer science, electrical networks, interconnected networks, biological networks, chemistry, etc.

The chemical graph theory (CGT) is a fast-growing area among researchers. It helps in understanding the structural properties of a molecular graph. There are many chemical compounds that possess a variety of applications in the fields of commercial, industrial, and pharmaceutical chemistry and daily life and in the laboratory.

In a chemical graph, the vertices represent atoms and edges refer to the chemical bonds in the underlying chemical structure. A topological index is a numerical value that is computed mathematically from the molecular graph. It is associated with the chemical constitution indicating the correlation of the chemical structure with many physical and chemical properties and biological activities [1–3].

Let *G* be a simple and connected graph with *G* as a *u* and *v* of *G* is equal to the length of (number of edges in) the shortest path that connects *u* and *v*. *i*th neighbor vertex set of *u*. Especially, if *u*, and *G*. If *u* a *G*.

A unicyclic graph of order *n* is a connected graph with *n* vertices and *m* edges. It is well-known that every unicyclic graph has exactly one cycle. Let *n* vertices. As usual, let *n*, respectively.

Let *G*, each of whose distance is equal to *k*. The Wiener polarity index, denoted by

There is another important graph-based structure descriptor, called Wiener index, based on distances in a graph. The

The name Wiener polarity index is introduced by Harold Wiener [4] in 1947. In Ref. [4], Wiener used a linear formula of

where *c* are constants for a given isomeric group.

If *G*, then

In 1998, Lukovits and Linert [5] demonstrated quantitative structure-property relationships in a series of acyclic and cycle-containing hydrocarbons by using the Wiener polarity index. In 2002, Hosoya [6] found a physicochemical interpretation of *n* vertices, respectively. Deng [8] characterized the extremal Wiener polarity indices among all chemical trees of order *n*. Hou [9] determined the maximum Wiener polarity index of unicyclic graphs and characterized the corresponding extremal graphs. Lei [8] determined the extremal trees with the given degree sequence with respect to the Wiener polarity index. In a previous study [10], the authors obtained the first and second smallest Wiener polarity indexes of unicyclic graphs. In this paper, we determine the third smallest Wiener polarity index of unicyclic graphs. Moreover, all the corresponding extremal graphs are characterized.

## 2. The Third Smallest Wiener Polarity Index of Unicyclic Graphs

The girth *G* is the length of a shortest cycle in *G*. Let

A *G* is a vertex of *G* which is not a pendant vertex. Suppose *U* is a unicyclic graph with unique cycle *w* is a non-pendant vertex and there is exactly one path connecting *x*}.

**Lemma 2.1.** [10] Let

**FIGURE 1**. Graphs

**Lemma 2.2.** Let

**Proof.** Since

**Lemma 2.3** [10]. Suppose

**Lemma 2.4**. Let

**Proof.** Let

Case 1.

This implies that U is a unicyclic graph obtained by attaching

Obviously,

Case 2. *U* and

the equality holds if and only if

By combining the above arguments, the result follows.

**Lemma 2.5** *Let**. If**, then the third smallest Wiener polarity index**, the equality holds if and only if* *or**and**are shown in*Figure 3).

**Proof.** Let

Case 1.

This implies that *U* is a unicyclic graph obtained by attaching

Obviously,

Case 2.

*U* and

the equality holds if and only if *U*, by Lemma 2.2, we have

the equality holds if and only if

By combining the above arguments, the result follows.

**Lemma 2.6** *Let* *If* *then the third smallest Wiener polarity index* *the equality holds if and only if* *and* *are shown in*Figure 5).

**FIGURE 5**. Graphs

**Proof.** Let

Case 1.

This implies that *U* is a unicyclic graph obtained by attaching

If

If

If

If *U* and

Case 2. *U* and

By combining the above arguments, the result follows.

**Lemma 2.7** *Let* *and* *If* *then* *if**then* *.*

**Proof.** When

When *U* and

**Lemma 2.8** Let

**Proof.** If

If

By combining the above arguments, the result follows.

**Theorem 2.9**. Let

**Proof.** By Lemma 2.4–2.8, the result follows.

## 3. Conclusions

Chemical graph theory is an important area of research in mathematical chemistry which deals with topology of molecular structure such as the mathematical study of isomerism and the development of topological descriptors or indices. In this paper, we first introduce some useful graph transformations and determine the third smallest Wiener polarity index of unicyclic graphs. In addition, all the corresponding extremal graphs are characterized.

## Data Availability Statement

All datasets presented in this study are included in the article.

## Author Contributions

WF performed conceptualization. FC and HD were responsible for methodology. WF and MM wrote the original manuscript. WF and FC reviewed and edited the article.

## Funding

This work was supported by open project of Anhui University (No. KF2019A01), Natural Science Research Foundation of Department of Education of Anhui Province (No. KJ2019A0817, No. KJ2020A0061), and National Science Foundation of China under grant (No. 11601001).

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## References

1. Liu J-B, Wang C, Wang S, Wei B. Zagreb indices and multiplicative zagreb indices of eulerian graphs. *Bull. Malays. Math. Sci. Soc* (2019) **42**:67–78. doi:10.1007/s40840-017-0463-2

2. Liu J-B, Pan X-F, Hu F-T, Hu F-F. Asymptotic Laplacian-energy-like invariant of lattices. *Appl Math Comput* (2015) **253**:205–14. doi:10.1016/j.amc.2014.12.035

3. Liu J-B, Pan X-F. Minimizing Kirchhoff index among graphs with a given vertex bipartiteness. *Appl Math Comput* (2016) **291**:84–8. doi:10.1016/j.amc.2016.06.017

4. Wiener H. Structural determination of paraffin boiling points. *J Am Chem Soc* (1947) **69**:17–20. doi:10.1021/ja01193a005

5. Lukovits I, Linert W. Polarity-numbers of cycle-containing structures. *J Chem Inf Comput Sci* (1998) **38**:715–9. doi:10.1021/ci970122j

6. Hosoya H. Mathematical and chemical analysis of Wieners polarity number. In: Rouvray DH, King RB, editors *Topology in chemistry-discrete mathematics of molecules*. **Vol. 57**. Chichester: Horwood (2002).

7. Du W, Li X, Shi Y. Algorithms and extremal problem on Wiener polarity index. *MATCH Commun. Math. Comput. Chem* (2009) **62**:235–44.

8. Lei H, Li T, Shi Y, Wang H. Wiener polarity index and its generalization in trees. *MATCH Commun. Math. Comput. Chem* (2017) **78**:199–212.

9. Hou H, Liu B, Huang Y. The maximum Wiener polarity index of unicyclic graphs. *Appl Math Comput* (2011) **218**:10149–57. doi:10.1016/j.amc.2012.03.090

Keywords: wiener polarity index, minimum, unicyclic graph, extremal graph, electrical networks

Citation: Fang W, Ma M, Chen F and Dong H (2020) Third Smallest Wiener Polarity Index of Unicyclic Graphs. *Front. Phys.* 8:553261. doi: 10.3389/fphy.2020.553261

Received: 18 April 2020; Accepted: 09 September 2020;

Published: 27 October 2020.

Edited by:

Muhammad Javaid, University of Management and Technology, PakistanReviewed by:

YaJing Wang, North University of China, ChinaAkbar Ali, University of Hail, Saudi Arabia

Copyright © 2020 Fang, Ma, Chen and Dong. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Fuyuan Chen, chen_fuyuan@sina.com