On the extremal values of the eccentric distance sum of trees

Discrete Applied Mathematics 186 (2015) 199–206

Contents lists available at ScienceDirect

Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam

On the extremal values of the eccentric distance sum of trees Lianying Miao ∗ , Qianqiu Cao, Na Cui, Shiyou Pang School of Science, China University of Mining and Technology, Xuzhou, Jiangsu, 221008, PR China

article

info

Article history: Received 7 June 2014 Received in revised form 27 January 2015 Accepted 30 January 2015 Available online 25 February 2015 Keywords: The eccentric distance sum Maximum degree Domination number Independence number

abstract Let G be a simple connected graph. The eccentric distance sum (EDS) of G is defined as v∈V εG (v)DG (v), where εG (v) is the eccentricity of the vertex v and DG (v) = u∈V dG (u, v) is the sum of all distances from the vertex v . In this paper, the trees having the maximal EDS among n-vertex trees with maximum degree ∆ and among those with domination number 3 are characterized. The trees having the maximal or minimal EDS among n-vertex trees with independence number α and the trees having the maximal EDS among n-vertex trees with matching number m are also determined. © 2015 Elsevier B.V. All rights reserved.

d ξ (G) =

1. Introduction In this paper, all graphs G = (V , E ) are finite, simple and undirected. For x ∈ V , NG (x) (or N (x)) is the set of vertices adjacent to x, and the degree of x, denoted by dG (x) (or d(x)) is |NG (x)| (or |N (x)|). We call u a leaf if d(u) = 1. We use ∆(G) and δ(G)(or ∆ and δ ) to denote the maximum degree and minimum degree of G, respectively. A vertex of degree k is called a k-v ertex. If uv ∈ E, we call u (or v ) a neighbor of v (or u). We call v a pendent neighbor of u if uv ∈ E and d(v) = 1. For u, v ∈ V , the distance dG (u, v) is defined as the length of the shortest path between u and v in G. DG (v) denotes the sum of all distances from v. The eccentricity ε(v) of a vertex v is the maximum distance from v to any other vertex. The centre of a graph is a vertex of minimum eccentricity. For W ⊆ V , G − W denotes the graph obtained from G by deleting the vertices in W together with their incident edges. If W = {w}, we just write G − w for G − {w}. If U ⊆ V , then G[U ] denotes the graph on U whose edges are precisely the edges of G with both ends in U. Let Sn , Pn and Kn be a star, a path and a complete graph on n vertices respectively. We use l(P ) to denote the length of a path P. For a real number x, we use ⌊x⌋ to denote the greater integer no greater than x and use ⌈x⌉ to denote the least integer no less than x. A subset S of V is called a dominating set of G if for every vertex v ∈ V \ S, there exists a vertex u ∈ S such that v is adjacent to u. The domination number of G, denoted by γ (G), is defined as the minimum cardinality of dominating sets of G. For a connected graph G of order n, Ore [20] obtained that γ (G) ≤ 2n . A subset M of E (G) is called a matching of G if no two edges are adjacent in G. The matching number of G, denoted by α ′ (G), is defined as the maximum cardinality of matching sets of G. A subset S of V (G) is called an independent set of G if no two vertices from S are adjacent in G. The independence number of G, denoted by α(G), is defined as the maximum cardinality of independent sets of G. It is known that for a bipartite graph G of order n and with δ > 0, α ′ (G) + α(G) = n. All trees are bipartite, hence the equation holds for the theorems in Section 5. A single number that can be used to characterize some property of the graph of a molecule is called a topological index. The topological index is a graph theoretic property that is preserved by isomorphism. The chemical information derived through the topological index has been found useful in chemical documentation, isomer discrimination, structure–property

∗

Corresponding author. E-mail address: [email protected] (L. Miao).

http://dx.doi.org/10.1016/j.dam.2015.01.042 0166-218X/© 2015 Elsevier B.V. All rights reserved.

200

L. Miao et al. / Discrete Applied Mathematics 186 (2015) 199–206

Fig. 1. ρ transformation.

correlations, etc. [1]. The properties of many topological indices such as the Wiener index [3], the degree distance index, the eccentric connectivity index [21,4,9,15,16,19], the eccentric distance sum [10] and so on are established and used as molecular descriptors [11,12]. Some results on the domination number can be found in [2,5–7,22,23]. The graph invariant-eccentric distance sum (EDS) was introduced by Gupta, Singh and Madan [10], which was defined as

ξ d (G) =



εG (v)DG (v).

v∈V

The eccentric distance sum can also be defined as

ξ d (G) =



(εG (u) + εG (v))dG (u, v).

u,v∈V

Many researchers have studied the eccentric distance sum of trees. Yu, Feng and Ilić [24] characterized the trees with the minimal EDS among the n-vertex trees of a given diameter. Li, Zhang, Yu, Feng [18] identified the trees with the minimal and second minimal eccentric distance sums among the n-vertex trees with matching number q and characterized the extremal tree with the second minimal eccentric distance sum among the n-vertex trees of a given diameter. They also determined the trees with the third and fourth minimal eccentric distance sums among the n-vertex trees. Geng, Li and Zhang [8] characterize the trees with the minimal EDS among n-vertex trees with domination number γ , and determine the trees with the maximal EDS among n-vertex trees with domination number γ satisfying n = kγ , where k = 2, 3, 2n . They also identify the trees with the minimal and the maximal EDS among the n-vertex trees with k leaves. Other results about the EDS of graphs can also be found in [13,14,17]. In this paper, we continue to study the eccentric distance sum of trees. The trees having the maximal EDS among n-vertex trees with maximum degree ∆ and among those with domination number 3 are characterized. The trees having the maximal or minimal EDS among n-vertex trees with independence number α and the trees having the maximal EDS among n-vertex trees with matching number m are also determined. 2. Preliminaries In this paper, we will use two graph transformations A1 and A2 posed in [14] and [8], respectively. A1 : Let T be a tree of order n > 3 and e = uv be a nonpendent edge. Suppose that T − e = T1 ∪ T2 with u ∈ V (T1 ) and v ∈ V (T2 ). A new tree T0 is obtained by identifying the vertex u of T1 with vertex v of T2 and attaching a leaf to the u(= v). T0 is said to be obtained by running an edge-growing transformation of T (on edge e = uv ), or e.g.t of T (on edge e = uv ) for short. Let T be a tree and uv be a pendent edge with dT (v) = 1 and dT (u) ≥ 3. Suppose uw ∈ E (T ) and w ̸= v . Let T0 = T − {uw} + {vw}. Then T0 is said to be obtained by running converse of e.g.t of T on uw. Lemma 2.1 ([14]). Let T be a tree of order n > 3 and e = uv be a nonpendent edge of T . If T0 is a tree obtained from T by running one step of e.g.t (on edge e = uv ), then we have ξ d (T0 ) < ξ d (T ). A2 : Let T be an arbitrary tree rooted at a center vertex and let v be a vertex of degree m + 1(m ≥ 2). Suppose that w is adjacent to v with εT (v) ≥ εT (w) and that T1 , T2 , . . . , Tm are subtrees under v with root vertices v1 , v2 , . . . , vm where NT (v) = {w, v1 , v2 , . . . , vm−1 , vm } and Tm is actually a path. Let T ′ = T − {vv1 , vv2 , . . . , vvm−1 } + {wv1 , wv2 , . . . , wvm−1 }. We say that T ′ is a ρ transformation of T and denote it by T ′ = ρ(T , v). See Fig. 1. Lemma 2.2 ([14]). Let T ′ be a ρ transformation of T defined as above, one has ξ d (T ) ≥ ξ d (T ′ ). The equality holds if and only if εT (v) = εT (w) and T [S ] is one of the longest paths in T , where S = V (G0 ) ∪ V (Tm ) ∪ {v} and G0 is the graph obtained from T (or T ′ ) by deleting V (T1 ) ∪ V (T2 ) ∪ · · · ∪ V (Tm ) ∪ {v}. A spider is a tree with at most one vertex of degree more than 2, called the hub of the spider (if there is no vertex of degree more than two, then any vertex can be the hub). A leg of a spider is a path from the hub to one of its leaves. Let S (a1 , a2 , . . . , ak ) be a spider with k legs P 1 , P 2 , . . . , P k for which the lengths l(P i ) = ai (i = 1, 2, . . . , k). It holds that

L. Miao et al. / Discrete Applied Mathematics 186 (2015) 199–206

201

Fig. 2. An illustration of tree T and T ′ . i=1 ai = n − 1. Graph S (a1 , a2 , . . . , ak ) is called a balancedspider if |ai − aj | ≤ 1 for 1 ≤ i, j ≤ k. A broom T is a spider with k legs where the lengths of at least k − 1 legs are 1, that is if T = S (a1 , 1, . . . , 1).

k

Lemma 2.3 ([8]). Suppose that P = v0 v1 · · · vi · · · vr · · · vd is one of the longest paths contained in an n-vertex tree T with dT (v1 ) ≤ dT (vd−1 ) and r = min{i|dT (vi ) ≥ 3, i = 2, 3, . . . , d − 1}. Let T ′ = T − {vr u|u ∈ NT (vr ) \ {vr −1 , vr +1 }} + {v1 u|u ∈ NT (vr ) \ {vr −1 , vr +1 }}. Then we have ξ d (T ) < ξ d (T ′ ). See Fig. 2. The corona of two graphs G1 and G2 , introduced in [7], is a new graph G = G1 ◦ G2 obtained from one copy of G1 with |V (G1 )| copies of G2 where the ith vertex of G1 is adjacent to every vertex in the ith copy of G2 . For example, the corona G ◦ K1 is a graph obtained from attaching a leaf to each vertex of G. Let Tn,γ be the set of all n-vertex trees with domination number γ . Lemma 2.4 ([8]). Among all the trees from Tn, n , the tree P n ◦ K1 has the maximal EDS. 2

2

Lemma 2.5 ([8]). Among all the trees from Tn,⌈ n ⌉ with n > 4, the tree Pn has the maximal EDS. 3

Let Pl (a, b) be an n-vertex tree obtained by attaching a and b leaves to the two end vertices of Pl = v1 v2 · · · vl , (l ≥ 2), respectively. Here, a + b = n − l, a, b ≥ 1. l l Lemma 2.6 ([8]). ξ d (Pl (1, n − l − 1)) < ξ d (Pl (2, n − l − 2)) < · · · < ξ d (Pl (⌊ n− ⌋, ⌈ n− ⌉)). 2 2

3. The extremal EDS of trees with maximum degree ∆ In this section, we are to determine the trees having the maximal EDS among the n-vertex trees with maximum degree

∆. Let T n,∆ be the set of all n-vertex trees with maximum degree ∆. When ∆ = 2, T n,∆ = {Pn } and when ∆ = n − 1, T n,∆ = {Sn }. From now on we assume that 3 ≤ ∆ ≤ n − 2. Theorem 3.1. Among T n,∆ (3 ≤ ∆ ≤ n − 2), S (a1 , 1, . . . , 1) maximizes the EDS, where a1 = n − ∆. Proof. Let T be an n-vertex tree with maximum degree ∆ that maximizes the EDS and let w be a ∆-vertex of T . If one leaf v is adjacent to a vertex u(̸= w) with dT (u) ≥ 3, after running the converse of e.g.t on one edge uu′ (u′ ̸= v) of T , we get a new tree T0 ∈ T n,∆ . By Lemma 2.1, ξ d (T0 ) > ξ d (T ), which contradicts the choice of T . So every leaf of T is adjacent to a 2-vertex or w . If T is not a spider, there is a vertex u ̸= w such that d(u) ≥ 3 and the connected branch Tu of T − uv containing u is a spider, where v is the neighbor of u on the path connecting w and u in T . Let uv1 v2 · · · vs and uu1 u2 · · · ul be the shortest and longest legs of Tu , respectively. Let T1 be the subtree obtained from T by deleting V (Tu ). We consider the following two cases. Case 1. In T , at least one of the longest paths from vs to any other vertex does not pass v. In this case, εT (vi ) = i + l for i ∈ {1, 2, . . . , s}, εT (u) = l and εT (x) = dT (x, u) + l for x ∈ V (T1 ). Let T ′ = T − {uv} + {vvs } (see Fig. 3) . We now prove that ξ d (T ) < ξ d (T ′ ). First we prove that εT ′ (x) ≥ εT (x) for all x ∈ V (T ). For x ∈ V (Tu )\{v1 , v2 , . . . , vs }, dT (x, y) = dT ′ (x, y) if y ∈ V (Tu ) and dT (x, y) = dT ′ (x, y)− s if y ∈ V (T1 ). So εT ′ (x) ≥ εT (x) for x ∈ V (Tu ) \ {v1 , v2 , . . . , vs }. For i ∈ {1, 2, . . . , s}, dT (vi , y) = dT ′ (vi , y) if y ∈ V (Tu ). So εT ′ (vi ) ≥ dT ′ (vi , ul ) = dT (vi , ul ) = i + l = εT (vi ) for i ∈ {1, 2, . . . , s}. Furthermore εT ′ (vi ) = εT (vi ) for i ∈ {⌈ 2s ⌉, . . . , s}. For x ∈ V (T1 ), εT ′ (x) ≥ dT ′ (x, ul ) = dT (x, u) + s + l > dT (x, u) + l = εT (x). This proves that εT ′ (x) ≥ εT (x) for all x ∈ V (T ). Now we consider DT (x) and DT ′ (x) for x ∈ V (T ). For x ∈ V (Tu ) \ {v1 , v2 , . . . , vs }, DT ′ (x) = DT (x) + |V (T1 )|. For i ∈ {1, 2, . . . , s}, DT ′ (vi ) = DT (vi ) + (s − 2i)|V (T1 )|.

202

L. Miao et al. / Discrete Applied Mathematics 186 (2015) 199–206

Fig. 3. An illustration of tree T and T ′ .

Fig. 4. An illustration of tree T and T ′′ .

For x ∈ V (T1 ), DT ′ (x) ≥ DT (x) + sl +

ξ d (T ′ ) − ξ d (T ) =



s

i=0

(s − 2i) = DT (x) + sl.

[εT ′ (x)DT ′ (x) − εT (x)DT (x)]

x∈V (T )



≥ 

εT (vj )s|V (T1 )| +







εT (x)sl

x∈V (T1 )

εT (vi )(s − 2i)|V (T1 )| +

i∈{1,2,...,s}

j∈{1,2,...,l}

>

εT (vi )(s − 2i)|V (T1 )| +

i∈{1,2,...,s}

x∈V (Tu )\{v1 ,v2 ,...,vs }

≥



εT (x)s|V (T1 )| +



εT (x)sl

x∈V (T1 )

[εT (ul−s+i )s + εT (vi )(s − 2i)]|V (T1 )|.

i∈{⌈ 2s ⌉,...,s}

For i ∈ {⌈ 2s ⌉, . . . , s}, εT (ul−s+i ) ≥ εT (vi ) and s ≥ 2i − s ≥ 0. So

ξ (T ) < ξ (T ). d

d

′



i∈{⌈ 2s ⌉,...,s}

[εT (ul−s+i )s + εT (vi )(s − 2i)]|V (T1 )| ≥ 0. So

Case 2. In T , all of the longest paths from vs to any other vertex pass v. In this case, there is a vertex z ∈ V (T1 ) such that εT (vs ) = dT (vs , z ) = dT (vs , u) + dT (u, z ) and dT (u, z ) > l. Furthermore, εT (vi ) = dT (vi , z ) = dT (u, z ) + i for all i ∈ {1, 2, . . . , s} and εT (uj ) = dT (uj , z ) = dT (u, z ) + j for all j ∈ {1, 2, . . . , l}. Let T ′′ = T − {uu1 } + {u1 vs } (see Fig. 4). We now prove that ξ d (T ) < ξ d (T ′′ ). First we prove that εT (x) ≤ εT ′′ (x) for all x ∈ V (T ). Let T2 be the subtree obtained from T by deleting {v1 , v2 , . . . , vs } and {u1 , u2 , . . . , ul }. For x ∈ V (T2 ), dT (x, y) = dT ′′ (x, y) if y ∈ V (T2 ) ∪ {v1 , v2 , . . . , vs } and dT (x, y) = dT ′′ (x, y) − s if y ∈ {u1 , u2 , . . . , us }. So εT ′′ (x) ≥ εT (x) for x ∈ V (T2 ). For i ∈ {1, 2, . . . , s}, dT (vi , y) = dT ′′ (vi , y) if y ∈ V (T2 ) ∪ {v1 , v2 , . . . , vs }. By the assumption in this case, dT (vi , z ) = dT (vi , u) + dT (u, z ) = εT (vi ). So εT ′′ (vi ) ≥ dT ′′ (vi , z ) = dT (vi , z ) = εT (vi ), that is εT ′′ (vi ) ≥ εT (vi ) for all i ∈ {1, 2, . . . , s}. In fact, εT ′′ (vi ) = εT (vi ) for all i ∈ {⌈ 2s ⌉, . . . , s}. For j ∈ {1, 2, . . . , l}, dT ′′ (uj , z ) = dT (uj , z ) + s = εT (uj ) + s > εT (uj ). So εT ′′ (uj ) ≥ εT (uj ) for all j ∈ {1, 2, . . . , l}. This proves that εT ′′ (x) ≥ εT (x) for all x ∈ V (T ). Now we consider DT (x) and DT ′′ (x) for x ∈ V (T ). For j ∈ {1, 2, . . . , l}, dT ′′ (uj , y) = dT (uj , y)+ s if y ∈ V (T2 )\ {u}; dT ′′ (uj , y) = dT (uj , y) if y ∈ {u1 , u2 , . . . , ul }; dT ′′ (uj , u) = dT (uj , vs ), dT ′′ (uj , vs−i ) = dT (uj , vi ) for i = 1, 2, . . . , s − 1, and dT ′′ (uj , vs ) = dT (uj , u). So DT ′′ (uj ) = DT (uj ) + s(|V (T2 )| − 1) for j ∈ {1, 2, . . . , l}.

L. Miao et al. / Discrete Applied Mathematics 186 (2015) 199–206

203

Fig. 5. T ⋆ with n = 22 and ∆ = 4.

For i ∈ {1, 2, . . . , s}, dT ′′ (vi , y) = dT (vi , y) y ∈ V (T2 ) ∪ {v1 , v2 , . . . , vs }, dT ′′ (vi , uj ) = dT (vi , uj ) + s − 2i if j ∈ {1, 2, . . . , l}. So DT ′′ (vi ) = DT (vi ) + (s − 2i)l for i ∈ {1, 2, . . . , s}. For x ∈ V (T2 ), dT ′′ (x, y) = dT (x, y) if y ∈ V (T2 ) ∪ {v1 , v2 , . . . , vs }, dT ′′ (x, uj ) = dT (x, uj ) + s if j ∈ {1, 2, . . . , l}. So DT ′′ (x) = DT (x) + sl for x ∈ V (T2 ). Note that εT ′′ (vi ) = εT (vi ) for all i ∈ {⌈ 2s ⌉, . . . , s}.

ξ d (T ′′ ) − ξ d (T ) =



[εT ′′ (x)DT ′′ (x) − εT (x)DT (x)]

x∈V (T )

≥







εT (uj )s(|V (T2 )| − 1) +

j∈{1,2,...,l}

x∈V (T2 )

−



εT (x)sl +

εT (vi )(s − 2i)l

i∈{1,2,...,⌈ 2s ⌉−1}

εT (vi )(2i − s)l

i∈{⌈ 2s ⌉,...,s}

>



[εT (ul−s+i )s(|V (T2 )| − 1) − εT (vi )(2i − s)l].

i∈{⌈ 2s ⌉,...,s}

For i ∈ {⌈ 2s ⌉, . . . , s}, εT (ul−s+i ) ≥ εT (vi ). Since s ≥ 2i − s ≥ 0 and |V (T2 )| − 1 ≥ l, so



i∈{⌈ 2s ⌉,...,s}

[εT (ul−s+i )s(|V (T2 )| −

1) − εT (vi )(2i − s)l] ≥ 0. So ξ (T ) < ξ (T ). Combining Cases 1 and 2, we know that T is a spider. Let T = S (a1 , a2 , . . . , a∆ ) with ∆ legs P 1 , P 2 , . . . , P ∆ such that l(P i ) = ai and a1 ≥ a2 ≥ · · · ≥ a∆ and the hub of T is w . It is obvious that the subgraph induced by V (P 1 ) ∪ V (P 2 ) is the longest path of T . If a2 ≥ 2, let NT (w) = {w1 , w2 , . . . , w∆−1 , w∆ }, P 1 = ww1 u2 u3 · · · ua1 , P 2 = ww2 v2 v3 · · · va2 . Let T˜ = T − {ww3 , ww4 , . . . , ww∆−1 , ww∆ } + d

d

′′

{va2 −1 w3 , va2 −1 w4 , . . . , va2 −1 w∆−1 , va2 −1 w∆ }, then T˜ ∈ T n,∆ . By Lemma 2.3, ξ d (T˜ ) > ξ d (T ), a contradiction. So T is a broom.



In this section, we determine the trees having the maximal EDS among the n-vertex trees in T n,∆ . For the trees having the minimum EDS, we have the following conjecture. Conjecture 3.2. Among T n,∆ , T ⋆ minimizes the EDS, where T ⋆ is a Volkmann tree. When n = 22, ∆ = 4, T ⋆ is depicted in Fig. 5. 4. The extremal EDS of trees with domination number 3 In [8], the tree having the minimal EDS among n-vertex trees with domination number γ was characterized and the tree having the maximal EDS among n-vertex trees with domination number γ for n = kγ (k = 2, 3, 2n ) was also determined. In this section, we characterize the tree having the maximal EDS among n-vertex trees with domination number γ = 3. Let T ∈ Tn,3 . As stated in the Introduction, by Ore [20] γ ≤ 2n , so n ≥ 6. If n = 6, the tree P n ◦ K1 has the maximal EDS 2 by Lemma 2.4. If 7 ≤ n ≤ 9, Pn has the maximal EDS by Lemma 2.5. From now on, we assume that n ≥ 10. We have the following theorem. 7 Theorem 4.1. Among Tn,3 with n ≥ 10, P7 (⌊ n− ⌋, ⌈ n−2 7 ⌉) maximizes the EDS . 2

Proof. Assume that T ∈ Tn,3 has the maximal EDS and S = {w1 , w2 , w3 } is a dominating set of T . Now we show that the following claim holds. Claim. S = {w1 , w2 , w3 } is an independent set.

204

L. Miao et al. / Discrete Applied Mathematics 186 (2015) 199–206

a

b

c

e

d

f

Fig. 6. • denotes the vertex that has no other neighbors.

Proof of Claim. W.l.o.g., we assume that w1 w2 ∈ E (G), then there is a path P connecting w1 , w2 and w3 . We may assume that the two ends of P are w1 and w3 respectively. Then l(P ) ≤ 4 since γ (T ) = 3. So there are at least 5 vertices not on P and at least one of w1 , w2 and w3 has at least two pendent neighbors. If w1 has no pendent neighbors, then γ (T ) = 2, a contradiction. So w1 has at least one pendent neighbor. If w1 has at least two pendent neighbors or w2 has at least one pendent neighbor, after running the converse of e.g .t on the edge w1 w2 of T , we obtain a new tree T1 , which still belongs to Tn,3 . By Lemma 2.1, we have ξ d (T1 ) > ξ d (T ), which contradicts the choice of T . So assume that w1 has one pendent neighbor and w2 has no pendent neighbor. Then w3 has at least four pendent neighbors. If l(P ) ≤ 3, after running the converse of e.g .t on the edge w2 w3 (if l(P ) = 2)or w2′ w3 (if l(P ) = 3) of T , where w2′ is the neighbor of w3 on P, we obtain a new tree T1 , which still belongs to Tn,3 . By Lemma 2.1, we have ξ d (T1 ) > ξ d (T ), which contradicts the choice of T . If l(P ) = 4, let P = w1 w2 v1 v2 w3 . After running the converse of e.g .t on the edge v2 w3 of T , we obtain a new tree T2 such that {w1 , v1 , w3 } is a dominating set of T2 , that is T2 ∈ Tn,3 . By Lemma 2.1, we have ξ d (T2 ) > ξ d (T ), which contradicts the choice of T . This completes the proof of the claim. Let Pij be the path connecting wi and wj . Let T ′ be the subtree induced by P12 ∪ P23 ∪ P13 . Then T ′ is one of the following graphs (see Fig. 6). If T ′ is the graph described in (a), one of w1 , w2 and w3 , say w1 , has dT (w1 ) ≥ 3 since n ≥ 10. After running the converse of e.g .t on the edge ww1 of T , we obtain a new tree T1 such that T1 ∈ Tn,3 and ξ d (T1 ) > ξ d (T ) by Lemma 2.1, which contradicts the choice of T . If T ′ is the graph described in (b), at least one of w2 and w3 has pendent neighbors for otherwise γ (T ) = 2, a contradiction. If one of w2 and w3 , say w2 , has dT (w2 ) ≥ 3, after running the converse of e.g .t on the edge ww2 of T , we obtain a new tree T1 such that T1 ∈ Tn,3 and ξ d (T1 ) > ξ d (T ) by Lemma 2.1, which contradicts the choice of T . So dT (w2 ) = dT (w3 ) = 2. Let T ′ = T − {ww3 } + {w2 w3 }, then ξ d (T ′ ) > ξ d (T ) by Lemma 2.3, which contradicts the choice of T . If T ′ is the graph described in (d) or (e), then either w2 has dT (w2 ) ≥ 3 or one of w1 and w3 has at least two pendent neighbors since n ≥ 10. We run the converse of e.g .t on v1 w2 if it is the former case and run the converse of e.g .t on w1 v1 or w3 v2 if it is the latter case to get a new tree T1 such that T1 ∈ Tn,3 and ξ d (T1 ) > ξ d (T ) by Lemma 2.1, which contradicts the choice of T . If T ′ is the graph described in (c), we consider the following cases: C1. dT (w1 ) = 1. W.l.o.g., we assume that dT (w2 ) ≤ dT (w3 ). Let T ′ = T − {ww1 } + {v1 w1 }, then T ′ ∈ Tn,3 with a dominating set {v1 , w2 , w3 } and ξ d (T ′ ) > ξ d (T ) by Lemma 2.2, which contradicts the choice of T . C2. dT (w1 ) = 2 and one of w2 and w3 , say w2 , has dT (w2 ) = 1. Let T ′ = T − {wv1 } + {w1 v1 }, then T ′ ∈ Tn,3 and ξ d (T ′ ) > ξ d (T ) by Lemma 2.3, a contradiction. C3. dT (w1 ) ≥ 3 and either dT (w2 ) = 1 or dT (w3 ) = 1. After running the converse of e.g .t on the edge ww1 of T , we obtain a new tree T1 such that T1 ∈ Tn,3 ({w1 , v1 , w3 } or {w1 , v2 , w2 } is a dominating set of T1 ) and ξ d (T1 ) > ξ d (T ) by Lemma 2.1, a contradiction. C4. dT (w1 ) ≥ 2, dT (w2 ) ≥ 2 and dT (w3 ) ≥ 2. Without loss of generality, we assume that dT (w2 ) ≤ dT (w3 ). Let T1 = T − {ww1 } + {w1 w2 }, then ξ d (T1 ) > ξ d (T ) by Lemma 2.3. Let T2 = T1 − {w2 y|y ∈ NT1 (w2 ) \ {v1 , w1 }} + {w1 y|y ∈ NT1 (w2 ) \ {v1 , w1 }} if dT1 (w1 ) < dT1 (w3 ) or T2 = T1 − {w2 y|y ∈ NT1 (w2 ) \ {v1 , w1 }} + {w3 y|y ∈ NT1 (w2 ) \ {v1 , w1 }} if dT1 (w1 ) ≥ dT1 (w3 ), then ξ d (T2 ) > ξ d (T1 ) by Lemma 2.3. So ξ d (T2 ) > ξ d (T ). It is clear that T2 ∈ Tn,3 ({w, w1 , w3 } is a dominating set of T2 ), this is a contradiction. From the above discussion, we have that T ′ is the graph described in (f). Without loss of generality, we assume that dT (w1 ) ≤ dT (w3 ). If dT (w2 ) ≥ 3, let T1 = T − {w2 y|y ∈ NT (w2 ) \ {v2 , v3 }} + {w1 y|y ∈ NT (w2 ) \ {v2 , v3 }}, then T1 ∈ Tn,3 and ξ d (T1 ) > ξ d (T ) by Lemma 2.3. So T is of the type of P7 [a, b], where a + b = n − 7. By Lemma 2.6, T ∼ = P7 (⌊ n−2 7 ⌋, ⌈ n−2 7 ⌉). This proves Theorem 4.1.  In this section, the tree having the maximal EDS among n-vertex tree with domination number γ = 3 is characterized. For other cases, we have the following conjecture. Conjecture 4.2. Among Tn,γ with 4 ≤ γ ≤

n , 3

Pα (⌊ n−α ⌋, ⌈ n−α ⌉) maximizes the EDS , where α = 3(γ − 1) + 1. 2 2

L. Miao et al. / Discrete Applied Mathematics 186 (2015) 199–206

205

Fig. 7. Tree Tn,β .

Fig. 8. An illustration of tree T ′′ (setting v ′ = v0 ).

5. The extremal EDS of trees with independence number α or matching number m In this section, we are to determine the trees having the maximal and minimal EDS among the n-vertex trees with independence number α . As a consequence, we determine the trees having the maximal EDS among the n-vertex trees with matching number m. Lemma 5.1 ([18]). Among all the trees of order n and with the matching number β, the tree Tn,β has the minimal EDS, and ξ d (Tn,β ) = 6n2 + β 2 + 9β n − 22n − 28β + 34 (see Fig. 7). By Lemma 5.1 and the fact that α ′ (T ) + α(T ) = n, the following theorem holds. Theorem 5.2. Among all the trees of order n and with the independence number α , the tree Tn,n−α has the minimal EDS. By Lemma 2.3, the following lemma holds. Lemma 5.3. Suppose that P = v0 v1 · · · vi · · · vr · · · vd is one of the longest paths contained in an n-vertex tree T with dT (v1 ) ≤ dT (vd−1 ) and r = min{i|dT (vi ) ≥ 3, i = 2, 3, . . . , d − 1}. Let v ′ be an adjacent vertex of v1 other than v2 and T ′′ = T − {vr u|u ∈ NT (vr ) \ {vr −1 , vr +1 }} + {v ′ u|u ∈ NT (vr ) \ {vr −1 , vr +1 }}. Then we have ξ d (T ) < ξ d (T ′′ ). In fact, T ′ in Lemma 2.3 is a tree obtained from T ′′ in Lemma 5.3 by running one step of e.g.t (on edge e = v1 v ′ ). See Figs. 2 and 8. Theorem 5.4. Among all the trees of order n and with the independence number α , the tree Pl (a, b) has the maximal EDS, where l = 2(n − α) − 1 and 0 ≤ b − a ≤ 1. Proof. Assume that T has the maximal EDS and S is a maximal independent set of T . Suppose that P = v0 v1 · · · vi · · · vr · · · vd is one of the longest paths contained in T with dT (v1 ) ≤ dT (vd−1 ). We can assume that (N (v1 ) \ v2 ) ∪ (N (vd−1 ) \ vd−2 ) ⊆ S. If there exists an i ∈ {2, 3, . . . , d − 2} such that dT (vi ) ≥ 3, let r = min{i|dT (vi ) ≥ 3, i = 2, 3, . . . , d − 2}. Let T ′ = T − {vr u|u ∈ NT (vr ) \ {vr −1 , vr +1 }} + {v1 u|u ∈ NT (vr ) \ {vr −1 , vr +1 }} if vr ̸∈ S and T ′ = T − {vr u|u ∈ NT (vr ) \ {vr −1 , vr +1 }} + {v ′ u|u ∈ NT (vr ) \ {vr −1 , vr +1 }} otherwise, where v ′ is an adjacent vertex of v1 other than v2 . It is easy to see that α(T ) = α(T ′ ). By Lemmas 2.2 and 5.3, ξ d (T ) < ξ d (T ′ ), which is a contradiction. So T ∼ = Pl (a, b) for some l. n −l l ∼ ⌋, ⌈ ⌉)) , by Lemma 2.6, T ( a , b ) with l = 2(n − α) − 1 P Since α(Pl (1, n − l − 1)) = α(Pl (2, n − l − 2)) = · · · = α(Pl (⌊ n− = l 2 2 and 0 ≤ b − a ≤ 1. This proves Theorem 5.4.  Theorem 5.5. Among all the trees of order n and with the matching number m, the tree Pl (a, b) has the maximal EDS, where l = 2m − 1 and 0 ≤ b − a ≤ 1. Acknowledgments The authors would like to thank the referees for their valuable corrections and suggestions on the manuscript that greatly improved its format and correctness. The first author was supported by National Natural Science Foundation of China (11271365).

206

L. Miao et al. / Discrete Applied Mathematics 186 (2015) 199–206

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

A.R. Ashrafi, M. Saheli, M. Ghorbani, The eccentric connectivity index of nanotubes and nanotori, J. Comput. Appl. Math. 235 (2011) 4561–4566. P. Dankelmann, W. Goddard, C.S. Swart, The average eccentricity of a graph and its subgraphs, Util. Math. 65 (2004) 41–51. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 (2001) 211–249. H. Dureja, S. Gupta, A.K. Madan, Predicting anti-HIV-1 activity of 6-arylbenzonitriles: computational approach using superaugmented eccentric connectivity topochemical indices, J. Mol. Graphics 26 (2008) 1020–1029. R.C. Entringer, D.E. Jackson, D.A. Snyder, Distance in graphs, Czechoslovak Math. J. 26 (1976) 283–296. J.F. Fink, M.S. Jocobson, L.K. Kinch, J. Roberts, On graphs having domination number half their order, Period. Math. Hungar. 16 (1985) 287–293. R. Frucht, F. Harary, On the corona of two graphs, Aequationes Math. 4 (1970) 322–324. X.Y. Geng, S.C. Li, M. Zhang, Extremal values on the eccentric distance sum of trees, Discrete Appl. Math. 161 (2013) 2427–2439. S. Gupta, M. Singh, A.K. Madan, Application of graph theory: relationship of eccentric connectivity index and Wiener index with anti-inflammatory activity, J. Math. Anal. Appl. 266 (2002) 259–268. S. Gupta, M. Singh, A.K. Madan, Eccentric distance sum: a novel graph invariant for predicting biological and physical properties, J. Math. Anal. Appl. 275 (2002) 386–401. I. Gutman, A property of the Wiener number and its modifications, Indian J. Chem. 36A (1997) 128–132. I. Gutman, W. Linert, I. Lukovits, A.A. Dobrynin, Trees with extremal hyper-Wiener index: mathematical basis and chemical applications, J. Chem. Inf. Comput. Sci. 37 (1997) 349–354. H.B. Hua, K.X. Xu, W.N. Shu, A short and unified proof of Yu et al.’s two results on the eccentric distance sum, J. Math. Anal. Appl. 382 (2011) 364–366. H.B. Hua, S.G. Zhang, K.X. Xu, Further results on the eccentric distance sum, Discrete Appl. Math. 160 (2012) 170–180. A. Ilić, Eccentric connectivity index, in: I. Gutman, B. Furtula (Eds.), Novel Molecular Structure Descriptors, Theory and Applications II, in: Math. Chem. Monogr., vol. 9, University of Kragujevac, 2010. A. Ilić, I. Gutman, Eccentric connectivity index of chemical trees, MATCH Commun. Math. Comput. Chem. 65 (2011) 731–744. A. Ilić, G.H. Yu, L.H. Feng, On the eccentric distance sum of graphs, J. Math. Anal. Appl. 381 (2011) 590–600. S.C. Li, M. Zhang, G.H. Yu, L.H. Feng, On the extremal values of the eccentric distance sum of trees, J. Math. Anal. Appl. 390 (2012) 99–112. M.J. Morgan, S. Mukwembi, H.C. Swart, On the eccentric connectivity index of a graph, Discrete Math. 311 (2011) 1229–1234. O. Ore, Theory of Graphs, in: Amer. Math. Soc. Colloq. Publ, vol. 38, 1962. V. Sharma, R. Goswami, A.K. Madan, Eccentric connectivity index: a novel highly discriminating topological descriptor for structure property and structure activity studies, J. Chem. Inf. Comput. Sci. 37 (1997) 273–282. B. Xu, E.J. Cockayne, T.W. Haynes, S.T. Hedetniemi, S.C. Zhou, Extremal graphs for inequalities involving domination parameters, Discrete Math. 216 (2000) 1–10. K.X. Xu, L.H. Feng, Extremal energies of trees with a given domination number, Linear Algebra Appl. 435 (2011) 2382–2393. G.H. Yu, L.H. Feng, A. Ilić, On the eccentric distance sum of trees and unicyclic graphs, J. Math. Anal. Appl. 375 (2011) 99–107.