Abstract
The Wiener polarity index of a graph G is the number of unordered pairs of vertices 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 [1–3].
Let G be a simple and connected graph with and . Sometimes we refer to G as a graph. For any , the 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. is called the ith neighbor vertex set of u. Especially, if , then (or for short) be the of u, and is called the of G. If , then we call u a 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 denote the class of unicyclic graphs on n vertices. As usual, let , , and be the star, cycle, and path of order n, respectively.
Let denote the number of unordered vertices pairs of G, each of whose distance is equal to k. The Wiener polarity index, denoted by , is defined to be the number of unordered vertices pairs of distance 3, i.e., .
There is another important graph-based structure descriptor, called Wiener index, based on distances in a graph. The is denoted by [4]
The name Wiener polarity index is introduced by Harold Wiener [4] in 1947. In Ref. [4], Wiener used a linear formula of and to calculate the boiling points of the paraffins, i.e.,where and c are constants for a given isomeric group.
If are the connected components of a graph G, then . 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 . 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 of a connected graph G is the length of a shortest cycle in G. Let be the unicyclic graph obtained from by adding one edge to two pendant vertices of .
A of G is a vertex of G which is not a pendant vertex. Suppose U is a unicyclic graph with unique cycle , in the sequel, we agree that and . For , let , where w is a non-pendant vertex and there is exactly one path connecting with x}.
Lemma 2.1. [10] Let , then , where equality holds if and only if is shown in Figure 1).
FIGURE 1
Lemma 2.2. Let and for any . be the new graph obtained from G by adding one vertex w and one edge adjacent to u in G. Then,
Proof. Since and , then
Lemma 2.3 [10]. Suppose . If and , then , where equality holds if and only if ( is shown in Figure 1).
Lemma 2.4. Let . If , then the third smallest Wiener polarity index , the equality holds if and only if , and are shown in Figure 1.
Proof. Let ; we consider the next cases.
Case 1.
This implies that U is a unicyclic graph obtained by attaching pendant vertices to , where . Without loss of generality, let . The graph is shown in Figure 2; by the definition of Wiener polarity index, we have
FIGURE 2
Obviously, the equality holds if and only if . Then the third smallest Wiener polarity index is .
Case 2. is the subgraph of U and ,, the equality holds if and only if by Lemma 2.2; we havethe equality holds if and only if or .
By combining the above arguments, the result follows.
Lemma 2.5Let. If, then the third smallest Wiener polarity index, the equality holds if and only iforandare shown inFigure 3).
FIGURE 3
Proof. Let , we consider the next cases.
Case 1.
This implies that U is a unicyclic graph obtained by attaching pendant vertices to , where . Without loss of generality, let . The graph is shown in Figure 4; by the definition of Wiener polarity index, we haveObviously, ; the equality holds if and only if . Then the third smallest Wiener polarity index is .
FIGURE 4
Case 2. and is the subgraph of U and and , by Lemma 2.2, we havethe equality holds if and only if . If is the induced subgraph of U, by Lemma 2.2, we havethe equality holds if and only if .
By combining the above arguments, the result follows.
Lemma 2.6Let. If, then the third smallest Wiener polarity index, the equality holds if and only if,, andare shown inFigure 5).
FIGURE 5
Proof. Let , we consider the next cases.
Case 1.
This implies that U is a unicyclic graph obtained by attaching pendant vertices to , where .
If , then there exists only one graph and .
If , then there exists only one graph and .
If , then there exists three graphs , , and , , .
If , then or or is the subgraph of U and . By Lemma 2.2, we have .
Case 2. is the subgraph of U and ; meanwhile, , the equality holds if and only if . By Lemma 2.2, we have , the equality holds if and only if .
By combining the above arguments, the result follows.
Lemma 2.7Letand. If, then; if, then.
Proof. When and , then there exists only one graph and .
When , is the subgraph of U and , by Lemma 2.2, we have .
Lemma 2.8 Let , if , then , the equality holds if and only if .
Proof. If , then by the definition of Wiener polarity index, we have .
If , then is the subgraph of U and . By Lemma 2.2, we have .
By combining the above arguments, the result follows.
Theorem 2.9. Let ; then the third smallest Wiener polarity index , the equality holds if and only if or ,, and are shown in Figure 1; and are shown in Figure 3; ,, and 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.
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).
Statements
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.
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.
LiuJ-BWangCWangSWeiB. Zagreb indices and multiplicative zagreb indices of eulerian graphs. Bull. Malays. Math. Sci. Soc (2019). 42:67–78. 10.1007/s40840-017-0463-2CrossRef Full Text | Google Scholar
2.
LiuJ-BPanX-FHuF-THuF-F. Asymptotic Laplacian-energy-like invariant of lattices. Appl Math Comput (2015). 253:205–14. 10.1016/j.amc.2014.12.035CrossRef Full Text | Google Scholar
3.
LiuJ-BPanX-F. Minimizing Kirchhoff index among graphs with a given vertex bipartiteness. Appl Math Comput (2016). 291:84–8. 10.1016/j.amc.2016.06.017CrossRef Full Text | Google Scholar
4.
WienerH. Structural determination of paraffin boiling points. J Am Chem Soc (1947). 69:17–20. 10.1021/ja01193a005CrossRef Full Text | Google Scholar
5.
LukovitsILinertW. Polarity-numbers of cycle-containing structures. J Chem Inf Comput Sci (1998). 38:715–9. 10.1021/ci970122jCrossRef Full Text | Google Scholar
6.
HosoyaH. Mathematical and chemical analysis of Wieners polarity number. In: DHRouvrayRBKing, editors Topology in chemistry-discrete mathematics of molecules. Vol. 57. Chichester: Horwood (2002). Google Scholar
7.
DuWLiXShiY. Algorithms and extremal problem on Wiener polarity index. MATCH Commun. Math. Comput. Chem (2009). 62:235–44. Google Scholar
8.
LeiHLiTShiYWangH. Wiener polarity index and its generalization in trees. MATCH Commun. Math. Comput. Chem (2017). 78:199–212. Google Scholar
9.
HouHLiuBHuangY. The maximum Wiener polarity index of unicyclic graphs. Appl Math Comput (2011). 218:10149–57. 10.1016/j.amc.2012.03.090Google Scholar
10.
LiuMLiuB. On the wiener polarity index. MATCH Commum. Math. Comput. Chem (2011). 66:293–304. Google Scholar
Summary
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
Volume
8 - 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
Updates
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
This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics
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