Abstract
The Wiener polarity index of a graph , usually denoted by , is defined as the number of unordered pairs of those vertices of that are at distance 3. A vertex of a tree with degree at least 3 is called a branching vertex. A segment of a tree is a nontrivial path whose end-vertices have degrees different from 2 in and every other vertex (if exists) of has degree 2 in . In this note, the best possible sharp lower bounds on the Wiener polarity index are derived for the trees of fixed order and with a given number of branching vertices or segments, and all the trees attaining this lower bound are characterized.
1. Introduction
A topological index is a numerical quantity calculated from a graph, which remains unchanged under graph isomorphism [1]. Topological indices have attracted much attention in recent years, as many of them provide a good correlation between the molecular structure of a chemical compound and its properties. Examples for calculating the topological indices of particular graphs can be found in [2ā4].
The Wiener polarity index is one of the oldest topological indices, which was proposed in 1947 by the chemist Harold Wiener [5], for predicting the boiling points of paraffins. The index for a graph is defined as the number of unordered pairs of those vertices of that are at distance 3. In the previous decade, has attracted much attention from researchers; for example, see the surveys [6, 7], papers [8ā25], and related references therein.
Before moving further, let us recall some definitions and notations first. All the graphs considered in this note are simple and finite. Let be a graph with the vertex set and the set of edges . The degree of a vertex is denoted by (or simply by if the graph under consideration is clear). The number of vertices in a graph is known as its order. A graph of order is called an -vertex graph. A vertex of degree 1 is called pendent vertex, while a vertex of degree greater than 2 is known as a branching vertex. Let (or ) be the set of all those vertices of that are adjacent to the vertex . As usual, we denote by and the path and the star graph of order , respectively. A segment of a tree is a nontrivial path (that is, a path of length at least 1) in with the property that both the end-vertices of have degrees different from 2 in and every other vertex (if exists) of has degree 2. A tree is called starlike tree (or generalized star) if it contains exactly one branching vertex (we call it the central vertex of ). A path in a tree is called a pendent path (internal path, respectively) of length , if one of the two vertices , is pendent and the other is branching (both the vertices and are branching, respectively) and if . The notation and terminology of (chemical) graph theory that are not defined in this note can be found in [1, 26ā28].
By using the definition of the Wiener polarity index, Lukovits and Linert [29] demonstrated the quantitative structure-property relationships in a series of acyclic and cycle-containing hydrocarbons. Considerable work has been done, however, on characterizing the trees that maximize or minimize under various additional conditions: for example, with given order [15], degree sequence [30, 31], diameter [32], and pendent vertices [33, 34]. Shafique and Ali [35] gave some structural properties of the trees of fixed order and with a given number of segments or branching vertices having maximum/minimum value. Here, in this note, we are specifically interested in extending the results obtained in the paper [35].
Du et al. [15] showed that of a tree can be written aswhere is the edge connecting the vertices . Here, it is important to note that coincides with reduced second Zagreb index [35ā37], for the case of trees.
For fixed integers and , denote by and the classes of all -vertex trees with segments and branching vertices, respectively, where and . In this note, we characterize all the trees attaining minimum value from each of the two classes and and hence provide the solution of a problem, left open in [35], concerning the minimum value.
Let be a tree obtained from a tree after applying a transformation such that . Throughout this note, whenever we consider such trees, by and we mean the degree and set of neighbors, respectively, of the vertex in .
2. Sharp Lower Bound on Wiener Polarity Index for -Vertex Trees with a Fixed Number of Segments
Note that consists of only the path graph , and is empty. Thus, we proceed in this note with the assumption . Denote by the starlike tree with pendent paths of length 1 (see Figure 1). Let be the class of all -vertex trees with exactly one internal path and pendent paths of length 1. For the tree(s) having the minimum Wiener Polarity index among all the members of the class , we firstly prove some lemmas.
Lemma 1. Let and be positive integers such that . If is a tree such that is minimum among all the trees of , then contains at most one pendent path of length greater than 1.
Proof. Suppose, contrarily, that and are two pendent paths in , where and (note that the vertices and may coincide). If , then , and we havea contradiction to the choice of .
Lemma 1 ensures that the trees and have the minimum value in the classes and , respectively. Also, it is obvious that the star graph gives the minimum value (that is, 0) in the class . Therefore, we proceed with the assumption . Denote by the subclass consisting of all starlike trees. Moreover, by Lemma 1, attains the minimum value in the class . Now, we consider the class where .
Lemma 2. Let and be positive integers such that . If is a tree having minimum value among all the members of , then each pendent path of is of length 1.
Proof. We contrarily assume that there is a pendent path of length in , where and . Let be a branching vertex different from and let be the neighbor of lying on the ā-ā path. Note that and that may coincide with . Let , it can be observed that , and we have a contradiction to the choice of .
Theorem 1. Let and be positive integers such that . If , thenand the equality sign in (3) holds if and only if either (see Figure 1) or .
Proof. If contains more than one pendent path of length at least 2, then by the proof of Lemma 1, there exists a tree having at most one pendent path of length at least 2 such that . Thus, it is enough to prove the result when contains at most one pendent path of length at least 2. In the remaining proof, we assume that has at most one pendent path of length at least 2.
If either or , then by elementary calculations, one has . We apply induction on to prove the desired result. Note that if or 4, then by Lemma 1, it holds that with equality if and only if . Also, if , then by using Lemmas 1 and 2, we have with equality if and only if either or . Next, suppose that and that the result holds for every satisfying .
Let be a longest path in , where . Note that each of the two vertices and has exactly one nonpendent neighbor in . Since contains at most one pendent path of length at least 2, at least one of the two vertices and is branching. Without loss of generality, we assume that is branching. Let where and for every . Let . Note that when , and when . Hence, by using the inductive hypothesis, we haveIf , then the equality holds if and only if and either or . If , then the equality holds if and only if and (because in this case, the tree contains a pendent path of length at least 2). Thus, we conclude that with equality if and only if or . This completes the induction and hence the proof.
3. Sharp Lower Bound on Wiener Polarity Index for -Vertex Trees with a Given Number of Branching Vertices
Recall that is the class of all -vertex trees with branching vertices, where . For , the star graph attains the minimum value (see [36]). Thus, throughout this section, we assume . Note that Lemma 3 may be proved in a fully analogous way to that of Lemma 2.
Lemma 3 (see [35]). Let and be positive integers such that . If is a tree having minimum value among all the members of , then every pendent path of is of length 1.
Let be the number of edges in a tree connecting the vertices of degrees and .
Lemma 4. Let and be positive integers such that . If is a tree having minimum value among all the members of and for some , then does not contain any pair of adjacent branching vertices.
Proof. Contrarily, suppose that is a pair of adjacent branching vertices and let be a pendent vertex adjacent to a vertex of degree at least 4. Note that may coincide with either of the vertices and . If , then it can be observed that , and we havewhich is positive because of the fact that the function is strictly increasing in both and where . Thus, we arrived at a contradiction to the choice of .
Lemma 5. Let and be positive integers such that . If is a tree with minimum among the trees from , such that with and , then a tree can be obtained from as , where is a nonpendent neighbor of , such that .
Proof. It holds, as it is easy to see that . Also, using the facts and , we havewhich implies .
Lemma 6. Let and be positive integers such that . If is a tree having minimum value among all the members of , then every vertex of degree greater than 3 in has exactly one nonpendent neighbor.
Proof. We contrarily assume that the vertex , with , has at least two nonpendent neighbors where . We consider the following cases:
Case 1. The vertex has at least one pendent neighbor.
Without loss of generality, we assume that for and for . Then, because has at least two nonpendent neighbors. Lemma 4 ensures that for every satisfying . If , then and hence, because of the fact , we havewhich is a contradiction.
Case 2. The vertex has nonpendent neighbor.
In this case, we have for every satisfying . Here, Lemmas 3ā5 ensure that there is a pendent vertex having the neighbor such that for , where . Let be the neighbor of that lies on the unique ā-ā path. If , then , and we havewhich is again a contradiction to the choice of .
Theorem 2. Let and be positive integers such that . If , thenand the equality holds if and only if , for , where : is a tree whose every vertex with degree has exactly one nonpendent neighbor and each internal path is of length at least 2}, and , for , where is a class of trees with degree sequence such that each pendent vertex of is adjacent to some branching vertex only.
Proof. Denote by the number of vertices of degree in a graph . Let be a tree that minimizes among the class . Lemma 3 and Lemma 4 conclude that whenever , every branching vertex in has degree 3 such that the vertices of degree 2 are placed between the adjacent vertices of degree 3 in such a way that no two vertices of degree 2 are adjacent if there are adjacent vertices of degree 3. Note that, for , we have , and . Hence, .
Now, Lemmas 3ā6 conclude that every internal path has a length of at least 2. Also, Lemma 5 ensures that, to obtain minimal graph , either we have to insert the vertices of degree 2 between any vertex of degree 2 and vertex of degree 3, or we have to add a starlike pendent vertex in such a way that every vertex with degree has exactly one nonpendent neighbor that is . Hence, , for , which completes the proof.
Data Availability
The data used to support the findings of the study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.