A Note on the Minimum Wiener Polarity Index of Trees with a Given Number of Vertices and Segments or Branching Vertices

'eWiener polarity index of a graph G, usually denoted by Wp(G), is defined as the number of unordered pairs of those vertices of G 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 T is a nontrivial path S whose end-vertices have degrees different from 2 in T and every other vertex (if exists) of S has degree 2 in T. In this note, the best possible sharp lower bounds on the Wiener polarity index Wp 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.


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 [2][3][4]. e Wiener polarity index W p 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. e index W p for a graph G is defined as the number of unordered pairs of those vertices of G that are at distance 3. In the previous decade, W p has attracted much attention from researchers; for example, see the surveys [6,7], papers [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], 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 G be a graph with the vertex set V(G) and the set of edges E(G). e degree of a vertex u ∈ V(G) is denoted by d u (G) (or simply by d u if the graph under consideration is clear). e number of vertices in a graph is known as its order. A graph of order n is called an n-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 N G (u) (or N(u)) be the set of all those vertices of G that are adjacent to the vertex u ∈ V(G). As usual, we denote by P n and S n the path and the star graph of order n, respectively. A segment S of a tree T is a nontrivial path (that is, a path of length at least 1) in T with the property that both the end-vertices of S have degrees different from 2 in T and every other vertex (if exists) of S has degree 2. A tree ST is called starlike tree (or generalized star) if it contains exactly one branching vertex (we call it the central vertex of ST ). A path P � v 0 , v 1 , . . . , v k in a tree T is called a pendent path (internal path, respectively) of length k, if one of the two vertices v 0 , v k is pendent and the other is branching (both the vertices v 0 and v k are branching, respectively) and e notation and terminology of (chemical) graph theory that are not defined in this note can be found in [1,[26][27][28].
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 W p 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 W p value. Here, in this note, we are specifically interested in extending the results obtained in the paper [35].
Du et al. [15] showed that W p of a tree T can be written as where uv is the edge connecting the vertices u, v ∈ V(T).
Here, it is important to note that W p coincides with reduced second Zagreb index [35][36][37], for the case of trees. For fixed integers n and s, denote by ST n,s and T n,b the classes of all n-vertex trees with s segments and b branching vertices, respectively, where 1 ≤ s ≤ n − 1 and 1 ≤ b ≤ (n/2) − 1. In this note, we characterize all the trees attaining minimum W p value from each of the two classes ST n,s and T n,b and hence provide the solution of a problem, left open in [35], concerning the minimum W p value.
Let T ′ be a tree obtained from a tree T after applying a transformation such that roughout this note, whenever we consider such trees, by d v and N(v) we mean the degree and set of neighbors, respectively, of the

Sharp Lower Bound on Wiener Polarity Index for n-Vertex Trees with a Fixed Number of Segments
Note that ST n,1 consists of only the path graph P n , and ST n,2 is empty. us, we proceed in this note with the assumption 3 ≤ s ≤ n − 1. Denote by S n s ∈ ST n,s the starlike tree with s − 1 pendent paths of length 1 (see Figure 1). Let ST * n,s ⊂ ST n,s be the class of all n-vertex trees with exactly one internal path and s − 1 pendent paths of length 1. For the tree(s) having the minimum Wiener Polarity index among all the members of the class ST n,s , we firstly prove some lemmas.

Lemma 1. Let n and s be positive integers such that
is minimum among all the trees of ST n,s , then T contains at most one pendent path of length greater than 1.
a contradiction to the choice of T. Lemma 1 ensures that the trees S n 3 and S n 4 have the minimum W p value in the classes ST n, 3 and ST n,4 , respectively. Also, it is obvious that the star graph S n gives the minimum W p value (that is, 0) in the class ST n,n−1 .
erefore, we proceed with the assumption 5 ≤ s ≤ n − 2. Denote by ST s ⊂ ST n,s the subclass consisting of all starlike trees. Moreover, by Lemma 1, S n s attains the minimum W p value in the class ST s . Now, we consider the class ST n,s ∖ST s where 5 ≤ s ≤ n − 2.
□ Lemma 2. Let n and s be positive integers such that 5 ≤ s ≤ n − 2. If T ∈ ST n,s ∖ST s is a tree having minimum W p value among all the members of ST n,s ∖ST s , then each pendent path of T is of length 1.
Proof. We contrarily assume that there is a pendent path P: Let v ∈ V(T) be a branching vertex different from v s and let u be the neighbor of v s lying on the v s -v path. Note that d u ≥ 2 and that u may coincide with v. Let and the equality sign in (3) holds if and only if either T � S n s (see Figure 1) or T ∈ ST * n,s .
Proof. If T ∈ ST n,s contains more than one pendent path of length at least 2, then by the proof of Lemma 1, there exists a tree T * having at most one pendent path of length at least 2 such that W p (T) > W p (T * ). us, it is enough to prove the result when T ∈ ST n,s contains at most one pendent path of length at least 2. In the remaining proof, we assume that T ∈ ST n,s has at most one pendent path of length at least 2.
If either T � S n s or T ∈ ST * n,s , then by elementary calculations, one has W p (T) � n − 3. We apply induction on s to prove the desired result. Note that if s � 3 or 4, then by Lemma 1, it holds that W p (T) ≥ n − 3 with equality if and only if T � S n s . Also, if s � 5, then by using Lemmas 1 and 2, we have W p (T) ≥ n − 3 with equality if and only if either T � S n 5 or T ∈ ST * n,5 . Next, suppose that 6 ≤ s ≤ n − 2 and that the result holds for every s ′ satisfying 3 ≤ s ′ ≤ s − 1.
Let P: w 1 , w 2 , . . . , w r be a longest path in T, where r ≥ 4. Note that each of the two vertices w 2 and w r−1 has exactly one nonpendent neighbor in T. Since T contains at most one pendent path of length at least 2, at least one of the two vertices w 2 and w r−1 is branching. Without loss of generality, we assume that w 2 is branching. Let N(w 2 ) � w 1 , w 3 , u 1 , u 2  T ′ � T − u 1 . Note that T ′ ∈ ST n−1,s−1 when t ≥ 2, and T ′ ∈ ST n−1,s−2 when t � 1. Hence, by using the inductive hypothesis, we have

Sharp Lower Bound on Wiener Polarity Index for n-Vertex Trees with a Given Number of Branching Vertices
Recall that T n,b is the class of all n-vertex trees with b branching vertices, where 1 ≤ b ≤ (n/2) − 1. For b � 1, the star graph S n attains the minimum W p value (see [36]). us, throughout this section, we assume 2 ≤ b ≤ (n/2) − 1. Note that Lemma 3 may be proved in a fully analogous way to that of Lemma 2.
Lemma 3 (see [35]). Let b and n be positive integers such that  Proof. Contrarily, suppose that w, z ∈ V(T) is a pair of adjacent branching vertices and let v ∈ V(T) be a pendent vertex adjacent to a vertex u ∈ V(T) of degree at least 4. Note that u may coincide with either of the vertices w and z. If then it can be observed that T ′ ∈ T n,b , and we have which is positive because of the fact that the function f(a, b) � ab − 2a − 2b + 3 is strictly increasing in both a and b where a, b ∈ (3, ∞]. us, we arrived at a contradiction to the choice of T. □ Lemma 5. Let b and n be positive integers such that 2 ≤ b ≤ (n/2) − 1. If T ∈ T n,b is a tree with minimum W p among the trees from T n,b , such that uv ∈ E(T) with d u � 1 and d v ≥ 4, then a tree T ′ ∈ T n,b can be obtained from T as Proof. It holds, as it is easy to see that T ′ ∈ T n,b . Also, using the facts d w ≥ 2 and d v ≥ 4, we have which implies W p (T) ≥ W p (T ′ ). □ Lemma 6. Let b and n be positive integers such that 2 ≤ b ≤ (n/2) − 1. If T ∈ T n,b is a tree having minimum W p value among all the members of T n,b , then every vertex of degree greater than 3 in T has exactly one nonpendent neighbor.
Proof. We contrarily assume that the vertex u ∈ V(T), with N(u) � u 1 , u 2 , . . . , u q , u q+1 , . . . , u t , has at least two nonpendent neighbors where t ≥ 4. We consider the following cases: e vertex u has at least one pendent neighbor. Without loss of generality, we assume that d u i � 1 for 1 ≤ i ≤ q and d u j � d j ≥ 2 for q + 1 ≤ j ≤ t.
en, t − q ≥ 2 because u has at least two nonpendent neighbors. Lemma 4 ensures that d j � 2 for every j satisfying q + 1 ≤ j ≤ t. If T ′ � T − uu t + u t u 1 , then T ′ ∈ T n,b and hence, because of the fact t − q ≥ 2, we have which is a contradiction.

Case 2.
e vertex u has nonpendent neighbor. In this case, we have d u i ≥ 2 for every i satisfying 1 ≤ i ≤ t. Here, Lemmas 3-5 ensure that there is a pendent vertex v ∈ V(T) having the neighbor w such that d w � 3 for d u i ≥ 2, where 1 ≤ i ≤ t. Let u 1 be the neighbor of u that lies on the unique vu path. If T ′ � T − uu t + vu t , then T ′ ∈ T n,b , and we have which is again a contradiction to the choice of T. 9) and the equality holds if and only if T ∈ T * 1 , for

Theorem 2. Let b and n be positive integers such that
T is a tree whose every vertex with degree ≥4 has exactly one nonpendent neighbor and each internal path is of length at least 2}, and T ∈ T * 2 , for Proof. Denote by N i the number of vertices of degree i in a graph G. Let T be a tree that minimizes W p among the class T n,b . Lemma 3 and Lemma 4 conclude that whenever (n − 1/3) ≤ b ≤ (n/2) − 1, every branching vertex in T 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 (n − 1/3) ≤ b ≤ (n/2) − 1, we have N 3 � b, N 1 � b + 2 and N 2 � n − 2b − 2. Hence, W p (T) � 4b − 4. Now, Lemmas 3-6 conclude that every internal path has a length of at least 2. Also, Lemma 5 ensures that, to obtain minimal graph T, 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 ≥4 has exactly one nonpendent neighbor that is T � T * 1 . Hence, W p (T) � n + b − 5, for 2 ≤ b < (n − 1/3), which completes the proof.

Data Availability
e data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.