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Front. Phys., 27 October 2020 | https://doi.org/10.3389/fphy.2020.553261

Third Smallest Wiener Polarity Index of Unicyclic Graphs

Wei Fang1,2, Muhan Ma3, FuYuan Chen4,5* and Hufeng Dong5
  • 1Anhui Province Key Laboratory of Animal Nutritional Regulation and Health, Anhui Science and Technology University, Fengyang, China
  • 2School of Mathematical Sciences, Anhui University, Hefei, China
  • 3Mathematics and Applied Mathematics, Reading Academy Nanjing University of Information Science and Technology, Nanjing, China
  • 4Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, China
  • 5Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu, China

The Wiener polarity index WP(G) of a graph G is the number of unordered pairs of vertices {u,v} where the distance between 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 [13].

Let G be a simple and connected graph with |V(G)|=n and |E(G)|=m. Sometimes we refer to G as a (n,m) graph. For any u,vV(G), the distancedG(u,v) between the vertices u and v of G is equal to the length of (number of edges in) the shortest path that connects u and v. NGi(u)={vV(G)|dG(u,v)=i} is called the ith neighbor vertex set of u. Especially, if i=1, then NG1(u)(or NG(u) for short) be the neighbor vertex set of u, and dG(u)=|NG(u)| is called the degree of G. If dG(u)=1, then we call u a pendant vertex of 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 Un denote the class of unicyclic graphs on n vertices. As usual, let K1,n1, Cn, and Pn be the star, cycle, and path of order n, respectively.

Let γ(G,k) denote the number of unordered vertices pairs of G, each of whose distance is equal to k. The Wiener polarity index, denoted by WP(G), is defined to be the number of unordered vertices pairs of distance 3, i.e., WP(G)=γ(G,3).

There is another important graph-based structure descriptor, called Wiener index, based on distances in a graph. The Wiener indexW(G) is denoted by [4]

W(G)=12u,vV(G)d(u,v)=k1γ(G,k).

The name Wiener polarity index is introduced by Harold Wiener [4] in 1947. In Ref. [4], Wiener used a linear formula of W(G) and WP(G) to calculate the boiling points tB of the paraffins, i.e.,

tB=aW(G)+bWP(G)+c,

where a,b, and c are constants for a given isomeric group.

If G1,,Gt are the connected components of a graph G, then WP(G)=i=1tWP(Gi). Therefore, it will suffice to consider the Wiener polarity index of connected graphs.

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 WP(G). Du et al. [7] obtained the smallest and largest Wiener polarity indices together with the corresponding graphs among all trees on 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(G) of a connected graph G is the length of a shortest cycle in G. Let S(n,1) be the unicyclic graph obtained from K1,n1 by adding one edge to two pendant vertices of K1,n1.

A nonpendant vertex of G is a vertex of G which is not a pendant vertex. Suppose U is a unicyclic graph with unique cycle Ct, in the sequel, we agree that V(Ct)={v1,v2,,vt} and E(Ct)={v1v2,v2v3,,vt1vt,v1vt}. For 1it, let li=max{d(vi,x), where w is a non-pendant vertex and there is exactly one path connecting vi with x}.

Lemma 2.1. [10] Let UUn, then WP(G)0, where equality holds if and only if US(n,1) or UC4 or UC5(S(n,1) is shown in Figure 1).

FIGURE 1
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FIGURE 1. Graphs S(n,1),S1,T1, T2, and T3.

Lemma 2.2. Let GUn and |NG2(u)|k for any uV(G). G+w be the new graph obtained from G by adding one vertex w and one edge adjacent to u in G. Then, WP(G+w)WP(G)+k.

Proof. Since NG+w(w)={u} and |NG2(u)|k, then WP(G+w)=WP(G)+|NG3(w)|=WP(G)+|NG2(u)|WP(G)+k.

Lemma 2.3 [10]. Suppose UUn\{S(n,1)}. If g(U)=3 and n5, then WP(U)n4, where equality holds if and only if US1 (S1 is shown in Figure 1).

Lemma 2.4. Let UUn. If g(U)=3, then the third smallest Wiener polarity index WP(U)=n3, the equality holds if and only if UTi,1i3(T1,T2, and T3 are shown in Figure 1).

Proof. Let C3={v1,v2,v3}; we consider the next cases.

Case 1. max{l1,l2,l3}=0.

This implies that U is a unicyclic graph obtained by attaching ki0 pendant vertices to vi, where 1i3. Without loss of generality, let k1+k3k2. The graph G1i(1i4) is shown in Figure 2; by the definition of Wiener polarity index, we have

WP(G11)=k1k2+k2k3+k1k3;WP(G12)=k1k2+k2k3;WP(G13)=(k1+k3+1)(k21)=k1k2+k2k3(k1+k3+1k2);WP(G14)=2(n5)n4(n6);WP(S1)=n4(n5).
FIGURE 2
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FIGURE 2. Graphs G1i(1i4).

Obviously, WP(G11)WP(G12)>WP(G13); the equality holds if and only if G11G12. Then the third smallest Wiener polarity index is WP(T1)=4=n3.

Case 2. max{l1,l2,l3}1.G15 is the subgraph of U and WP(G15)=2,|NG152(u)|1, the equality holds if and only if uv4 by Lemma 2.2; we have

WP(U)WP(G15)+n5=n3,

the equality holds if and only if T2 or T3.

By combining the above arguments, the result follows.

Lemma 2.5 LetUUn. Ifg(U)=4, then the third smallest Wiener polarity indexWP(U)=n3, the equality holds if and only if UT4orT5(T4andT5are shown inFigure 3).

FIGURE 3
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FIGURE 3. Graphs S2, T4, and T5.

Proof. Let C4={v1,v2,v3,v4}, we consider the next cases.

Case 1. max{l1,l2,l3,l4}=0.

This implies that U is a unicyclic graph obtained by attaching ki0 pendant vertices to vi, where 1i4. Without loss of generality, let k1+k3k2+k4. The graph G2i(1i4) is shown in Figure 4; by the definition of Wiener polarity index, we have

WP(G21)=k1k2+k2k3+k3k4+k1k4+i=14ki,WP(G22)=k1k2+k2k3+k3k4+k1k4+i=14ki,WP(G23)=k1k2+k2k3+k3k4+k1k4+i=14ki,WP(G24)=(k1+k3+1)(k2+k41)+i=14ki.

Obviously, WP(G21)=WP(G22)=WP(G23)>WP(G24)n3; the equality holds if and only if G24T4. Then the third smallest Wiener polarity index is WP(T4)=3=n3.

FIGURE 4
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FIGURE 4. Graphs G2i(1i4).

Case 2. max{l1,l2,l3,v4}1.

S2(k1=1 and k2=0) is the subgraph of U and WP(S2)=1(k1=1 and k2=0), by Lemma 2.2, we have

WP(U)1+n5=n4,

the equality holds if and only if US2(k1=1,k2=1). If S2(k1=1,k2=1) is the induced subgraph of U, by Lemma 2.2, we have

WP(U)2+n5=n3,

the equality holds if and only if UT5.

By combining the above arguments, the result follows.

Lemma 2.6 Let UUn. If g(U)=5, then the third smallest Wiener polarity index WP(U)=n3, the equality holds if and only if UTi, (i=6,7,8) (T6, T7, and T8are shown inFigure 5).

FIGURE 5
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FIGURE 5. Graphs S3,G31,T6,T7, and T8.

Proof. Let C5={v1,v2,v3,v4,v5}, we consider the next cases.

Case 1. max{l1,l2,l3,l4,l5}=0.

This implies that U is a unicyclic graph obtained by attaching ki0 pendant vertices to vi, where 1i5.

If n=5, then there exists only one graph C5 and WP(C5)=0.

If n=6, then there exists only one graph S3 and WP(S3)=2=n4.

If n=7, then there exists three graphs G31, T6, and T7, WP(G31)=5=n2, WP(T6)=WP(T7)=4=n3.

If n>7, then G31 or T6 or T7 is the subgraph of U and min{|NG312(u)|,|NT62(u)|,|NT72(u)|}2. By Lemma 2.2, we have WP(U)4+2+n8=n2.

Case 2. max{l1,l2,l3,l4,l5}1.T8(n=t=7) is the subgraph of U and WP(T8)=4(n=t=7); meanwhile, |NT82(u)|1, the equality holds if and only if u=v7. By Lemma 2.2, we have WP(U)4+n7=n3, the equality holds if and only if U=T8.

By combining the above arguments, the result follows.

Lemma 2.7 Let UUn and g(U)=6. If n=6, then WP(C6)=n3; ifn>7, then WP(U)n2.

Proof. When g(U)=6 and n=6, then there exists only one graph C6 and WP(C6)=3=n3.

When n7, C6 is the subgraph of U and |NC62(u)|2, by Lemma 2.2, we have WP(U)3+2+n7=n2.

Lemma 2.8 Let UUn, if g(U)=s7, then WP(U)n, the equality holds if and only if UCs.

Proof. If UCs, then by the definition of Wiener polarity index, we have WP(U)=n.

If UCs, then Cs(s7) is the subgraph of U and |NCs2(u)|2. By Lemma 2.2, we have WP(U)WP(Cs)+2+(ns1)=n+1.

By combining the above arguments, the result follows.

Theorem 2.9. Let UUn; then the third smallest Wiener polarity index WP(U)=n3, the equality holds if and only if UC6 or Ti,1i8 (T1,T2, and T3 are shown in Figure 1; T4 and T5 are shown in Figure 3; T6,T7, and T8 are shown in Figure 5).

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.

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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, Pakistan

Reviewed by:

YaJing Wang, North University of China, China
Akbar 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