# Extremal Wiener Index of Trees with All Degrees Odd

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1 MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) ISSN Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of Sciences, Jimei University, Xiamen, Fujian, , P. R. China (Received December 17, 2012) Abstract The Wiener index of a graph is defined as the sum of distances between all pairs of vertices of the graph. In this paper, we characterize the trees which maximize and minimize the Wiener index among all trees of given order that have only vertices of odd degrees. 1 Introduction All graphs considered in this paper are simple and connected. Let G be a graph with vertex set V (G) and edge set E(G). The degree d G (v) of a vertex v in G is the number of edges of G incident with v. The distance between vertices u and v of G is denote by d G (u, v). The Wiener index of a graph G is defined as W (G) = d G (u, v). (1) {u,v} V (G) The Wiener index belongs among the oldest graph-based structure descriptors (topological indices). It was first introduced by Wiener [13] and has been extensively studied in many literatures. Numerous of its chemical applications and mathematical properties This project is supported by the Natural Science Foundation of Fujian Province of China (No. 2010J01008) and FJCEF (JA11163)

2 -288- are well studied [1,3,6,8,9,11]. For detailed results on this topic, the readers may referred to [4,5]. Chemists are often interested in the Wiener index of certain trees which represent molecular structures. Let T be a tree and e = uv an edge of T. Wiener [13] found that the formula (1) is equal to W (T )= n T1 (e) n T2 (e) (2) e E(T ) where n T1 (e) (resp. n T2 (e)) is the number of vertices of the component of T e containing u (resp. v). Denote by K 1,n 1 and P n the star and the path with n vertices, respectively. Entringer et al. [6] proved the following result which bounds the Wiener index of a tree in term of its order. Theorem 1 ([6]). Let T be a tree on n vertices, then ( ) n +1 (n 1) 2 W (T ). 3 The lower bound is achieved if and only if T = K 1,n 1 and the upper bound is achieved if and only if T = P n. Since every atom has a certain valency, chemists are interested in trees with some restricted degree conditions, having maximal or minimal Wiener index. Fischermann et al. [7] determined the trees which have the minimum Wiener index among all trees of given order and maximum degree. If a graph G has vertices v 1, v 2,...,v n,thenthe sequence (d G (v 1 ),d G (v 2 ),...,d G (v n )) is called a degree sequence of G. It is well known [2] that a sequence (d 1,d 2,...,d n )ofpositiveintegersisadegreesequenceofann vertex n tree if and only if d n =2(n 1). A tree T is called a caterpillar if the tree obtained i=1 from T by removing all pendent vertices is a path. Shi [10] obtained the following result. Theorem 2 ([10]). Let (d 1,d 2,...,d n ) be a degree sequence with n d n =2(n 1), and T max be the tree with maximal Wiener index among all trees which have this particular degree sequence. Then T max is a caterpillar. Wang [12] and Zhang et al. [14] independently characterized the tree that minimizes the Wiener index among trees of given degree sequences. Recently, by using Theorem 2, Zhang et al. [15] characterized the tree that maximizes the Wiener index among trees of given degree sequences. i=1

3 -289- In this paper, we continue to study the extremal Wiener index of trees with specific degree conditions. It is well known that every nontrivial tree has at least two vertices of degree one. This motivates the central problem that we consider in this paper. Problem A. Which trees minimize or maximize the Wiener index among all trees of given order that have only vertices of odd degrees? Since the relation d G (v) =2 E(G) holds for any graph G, it implies that each v V (G) tree that has only vertices of odd degree must have even number of vertices. Let T 2n be the set of all trees on 2n vertices whose vertices are all of odd degrees. Let F 2n be the tree shown in Figure n F 2n Fig. 1. The tree with odd vertex degrees, having greatest Wiener index. The main result of this paper is as follows which settles Problem A. Theorem 3. Let T T 2n.Then W (K 1,2n 1 ) W (T ) W (F 2n ) the lower bound is achieved if and only if T = K 1,2n 1 and the upper bound is achieved if and only if T = F 2n. 2 Proof of Theorem 3 Proof. Note that K 1,2n 1 T 2n, thus by Theorem 1, we have W (T ) W (K 1,2n 1 )with equality if and only if T = K 1,2n 1. Now we turn to determine the upper bound of W (T ). Let T be a tree with maximal Wiener index in T 2n. Suppose (d 1,d 2,...,d 2n ) is the degree sequence of T.LetT d be the set of all trees with this degree sequence (d 1,d 2,...,d 2n ). Clearly T d is a subclass of T 2n, so T also is a tree with maximal Wiener index in T d. By Theorem 2, T is a caterpillar.

4 -290- Let P = y 0 y 1...y l y l+1 be a longest path in T. Assume that T F 2n, then there exists a vertex y i (1 i l) such that d T (y i )=2t Let y i 1, y i+1, u 1, u 2,..., u 2t 1 be the neighbors of y i. Now we are going to construct a tree T T such that T T 2n and W (T ) >W(T ). In order to do so, we delete the pendent vertex u 2t 1 and the edges y i u 1, y i u 2,..., y i u 2t 2 from T, split y i into two adjacent vertices y i and y i and further join u 1, u 2,..., u 2t 3 to y i and join u 2t 2 to y i. The resulting tree is denoted by T. See Figure 2 for an example. u 1 u 2 u 3 u 1 u 2 e 1 e 2 = f 1 f 2 f 3 y 0 y 1 y 2 y 3 y 4 y 0 y 1 y 2 y 2 y 3 y 4 T T Fig. 2. The trees used in the proof of Theorem 3. Obviously, T T 2n. Denote by e 1, e 2,..., e l 1 and f 1, f 2,..., f l the consecutive edges on the path P 1 = y 1...y i...y l of T and on the path P 2 = y 1...y iy i,...y l of T from y 1 to y l, respectively (see Figure 2). Note that T has E(T ) (l 1) = (2n 1) (l 1) = 2n l pendent edges and T has E(T ) l =(2n 1) l pendent edges. Now by the formula (2) we have W (T ) = = e E(T ) n T 1(e) n T 2(e) e is a pendent edge n T 1(e) n T 2(e)+ e is not a pendent edge n T 1(e) n T 2(e) and l 1 = (2n l)(2n 1) + n T 1(e k ) n T 2(e k ) (3) k=1 W (T ) = n T 1(e) n T 2(e) e E(T ) = n T 1(e) n T 2(e)+ n T 1(e) n T 2(e) e is a pendent edge e is not a pendent edge l = (2n l 1)(2n 1) + n T 1(f k ) n T 2(f k ). (4) k=1

5 -291- It is easily checked that n T 1(e j ) n T 2(e j )=n T 1(f j ) n T 2(f j )holdsforanyj {1, 2,..., i 1} and n T 1(e r ) n T 2(e r )=n T 1(f r+1 ) n T 2(f r+1 )holdsforanyr {i, i +1,...,l 1}. Now from (3) and (4), we arrive at W (T ) W (T ) = n T 1(f i ) n T 2(f i ) (2n 1) = n T 1(f i ) n T 2(f i ) n T 1(f i ) n T 2(f i ) + 1 (Since n T 1(f i )+n T 2(f i )=2n.) =[n T 1(f i ) 1][n T 2(f i ) 1] > 0. (Since n T 1(f i ) > 1,n T 2(f i ) > 1.) But this contradicts to the choice of T. While Problem A is settled, it is also natural to consider the analogous questions for general graphs with other degree restrictions. Denote by O 2n the set of graphs on 2n vertices whose vertices are all of odd degree. Denote by E n the set of connected graphs on n vertices whose vertices are all of even degree. An Euler tour of G is a closed walk that traverses each edge of G exactly once. A graph is Eulerian if it contains an Euler tour. It is well known [2] that E n is the set of all Eulerian graphs on n vertices. The following problem is worthwhile to study. Problem B. Characterize the graphs with maximal Wiener index in O 2n and in E n, respectively. L Fig. 3. A 10-vertex graph whose Wiener index exceeds the Wiener index of the tree F 10. Although T 2n O 2n and F 2n is the unique graph with maximal Wiener index in T 2n,weremarkthatF 2n may not be the graph with maximal Wiener index in O 2n.For an example, let L be the graph shown in Figure 3. Then L O 10. A straightforward calculation gives that W (L) = 125 >W(F 10 ) = 121. Settling Problem B seems to be difficult.

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