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On the extremal values of the eccentric distance sum of trees with a given domination number
Discrete Applied Mathematics (
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On the extremal values of the eccentric distance sum of trees with a given domination number Lianying Miao *, Shiyou Pang, Fang Liu, Eryan Wang, Xiaoqing Guo School of Science, China University of Mining and Technology, Xuzhou, Jiangsu, 221008, PR China
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Article history: Received 13 March 2016 Received in revised form 24 April 2017 Accepted 26 April 2017 Available online xxxx Keywords: Tree Eccentric distance sum Domination number
a b s t r a c t Let G be a∑ simple connected graph. The eccentric distance sum (EDS) of G is defined as d ξ∑ (G) = 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 extremal tree among n-vertex trees with domination number γ satisfying 4 ≤ γ < ⌈ 3n ⌉ having the maximal EDS is characterized. This proves Conjecture 4.2 posed in Miao et al. (2015). © 2017 Elsevier B.V. All rights reserved.
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 for short) to denote the maximum degree and minimum degree of G, respectively. For u, v ∈ V , the distance dG (u, v ) is defined as the length of a shortest path between u and v in G. DG (v ) denotes the sum of all distances from v . The eccentricity εG (v ) of a vertex v is the maximum distance from v to any other vertex. The diam(G) of a graph G is the maximum eccentricity of any vertex in G. 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}. For E ′ ⊆ E, G \ E ′ denotes the graph obtained from G by deleting the edges in E ′ . If E ′ = {e}, we just write G \ e for G − {e}. 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 Pn be a path on n vertices. We use l(P) to denote the length of a path P. For a real number x, we use ⌊x⌋ to denote the greatest integer no greater than x and use ⌈x⌉ to denote the least integer no less than x. 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. 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 [12] obtained that γ (G) ≤ 2n . Some results on the domination number can be found in [1,13,16]. The graph invariant-eccentric distance sum (EDS) was introduced by Gupta, Singh and Madan [3], which was defined as
ξ d (G) =
∑
εG (v )DG (v ).
v∈V
*
Corresponding author. E-mail address: [email protected] (L. Miao).
http://dx.doi.org/10.1016/j.dam.2017.04.032 0166-218X/© 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.
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Fig. 2.1. An illustration of trees T and T ′ in A2 .
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 [18] characterized the trees with the minimal EDS among the n-vertex trees of a given diameter. Li, Zhang, Yu, Feng [9] 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 trees with the second, third and fourth minimal eccentric distance sum among the n-vertex trees of a given diameter. Geng, Li and Zhang [2] and Miao [10] 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 and 3n . They also identify the trees with the minimal and the maximal EDS among the n-vertex trees with k leaves. In [10] the trees having the maximal EDS among n-vertex trees with maximum degree ∆ 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, respectively. Other results about the EDS of graphs can also be found in [4,5,7,8]. For other eccentricity-based graph invariants one may refer to [6,11,14,15,17,19,20]. In this paper, we continue to study the eccentric distance sum of trees. The extremal tree among n-vertex trees with domination number γ satisfying 4 ≤ γ < ⌈ 3n ⌉ having the maximal EDS is characterized. This proves Conjecture 4.2 posed in [10]. 2. Preliminaries The following graph transformations A1 and A2 were posed in [5] and [2] respectively. A1 : Let T be a tree of order n > 3 and e = uv be a nonpendant 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 ([5]). Let T be a tree of order n > 3 and e = uv be a nonpendant 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 : Suppose that P = v0 v1 · · · vi · · · vk is one of the longest paths contained in an n-vertex tree T with dT (v1 ) ≤ dT (vk−1 ) and i = min{j|dT (vj ) ≥ 3, j = 2, 3, . . ., k − 2}. Let T ′ = T − {vi u|u ∈ NT (vi ) \ {vi−1 , vi+1 }} + {v1 u|u ∈ NT (vi ) \ {vi−1 , vi+1 }}. Then T ′ is said to be obtained by running an A2 transformation of T . See Fig. 2.1. Lemma 2.2 ([2]). Let T be an n-vertex tree. If T ′ is a tree obtained from T by running an A2 , then we have ξ d (T ′ ) > ξ d (T ). Let Tn,γ be the set of all n-vertex trees with domination number γ . 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. Lemma 2.3 ([2]). Among all the trees from Tn,⌈ n ⌉ with n > 4, the tree Pn has the maximal EDS. 3
4 Lemma 2.4 ([2]). Among Tn,2 with n ≥ 4, P4 (⌊ n− ⌋, ⌈ n−2 4 ⌉) maximizes the EDS. 2 7 Lemma 2.5 ([10]). Among Tn,3 with n ≥ 10, P7 (⌊ n− ⌋, ⌈ n−2 7 ⌉) maximizes the EDS. 2
Lemma 2.6 ([2]). ξ d (Pl (1, n − l − 1)) < ξ d (Pl (2, n − l − 2)) < · · · < ξ d (Pl (⌊ n2−l ⌋, ⌈ n2−l ⌉)).
Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.
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Fig. 3.1. An illustration of trees T and T ′ in A3 .
3. The extremal EDS of n-vertex tree with domination number at most ⌈ n3 ⌉ By Lemmas 2.3–2.5, we assume that 4 ≤ γ < ⌈ 3n ⌉. Let T ∈ Tn,γ with 4 ≤ γ < ⌈ 3n ⌉. By 4 < ⌈ 3n ⌉, n ≥ 13. From now on, we assume that n ≥ 13. We have the following theorem. Theorem 3.1. Among Tn,γ (4 ≤ γ < ⌈ 3n ⌉), Pl (⌊ n2−l ⌋, ⌈ n2−l ⌉) maximizes the EDS, where l = 3γ − 2. Before the proof of Theorem 3.1, we introduce two graph transformations A3 and A4 as follows. A3 : Let T be an n-vertex tree of order n > 12 and P = v0 v1 · · · vk−1 vk be one of the longest paths in T with k ≥ 4. Let A = {x|x ∈ NT (v1 ) \ v2 }, B = {x|x ∈ NT (vk−1 ) \ vk−2 }, and assume that |A| = a ≤ |B| = b. Let i = min{j|dT (vj ) ≥ 3, 2 ≤ j ≤ k − 2}. Assume dT (vi ) ≥ 4 and w ∈ NT (vi ) \{vi−1 , vi+1 }. Let T ′ = T \{vi y|y ∈ NT (vi ) \{vi−1 , vi+1 , w}}+{v1 y|y ∈ NT (vi ) \{vi−1 , vi+1 , w}}. Then T ′ is said to be obtained by running an A3 transformation of T . See Fig. 3.1. Lemma 3.2. Let T be an n-vertex tree. If T ′ is a tree obtained from T by running an A3 , then we have ξ d (T ′ ) > ξ d (T ). Proof. In T , let T0 be the component of T \ vi containing w , T1 be the component of T \ vi containing vi+1 , T2 be the tree induced by A ∪ {v1 , v2 , . . . , vi } and F3 be the forest induced by the vertices not in V (T0 ) ∪ V (T1 ) ∪ V (T2 ). We let |V (F3 )| = f3 . First we consider εT (x) and εT ′ (x) for x ∈ V (T ). (I) If x ∈ V (T ) \ V (F3 ), εT ′ (x) ≥ max{dT ′ (x, v0 ), dT ′ (x, vk )} = max{dT (x, v0 ), dT (x, vk )} = εT (x). (II) If x ∈ V (F3 ), εT ′ (x) ≥ dT (v0 , vk ) = εT (v0 ) ≥ εT (x). Now we consider DT (x) and DT ′ (x) for x ∈ V (T ). (I′ ) If x ∈ V (T0 ) ∪ V (T1 ), dT ′ (x, y) = dT (x, y) for y ∈ V (T ) \ V (F3 ) and dT ′ (x, y) = dT (x, y) + i − 1 ≥ dT (x, y) for y ∈ V (F3 ). So DT ′ (x) = DT (x) + (i − 1)f3 > DT (x). (II′ ) If x ∈ V (F3 ), dT ′ (x, y) = dT (x, y) + (i − 1) for y ∈ V (T0 ) ∪ V (T1 ), dT ′ (x, y) = dT (x, y) for y ∈ V (F3 ), dT ′ (x, vj ) = dT (x, vj ) + 2j − (i + 1) for j ∈ {1, 2, . . . , i} and dT ′ (x, y) = dT (x, y) − (i − 1) for y ∈ A. So, by the fact that A ⊂ V (T2 ), B ⊂ V (T1 ) and a ≤ b,
∑
DT ′ (x) =
[dT (x, y) + (i − 1)] +
y∈V (T0 )∪V (T1 )
+
∑
dT (x, y)
y∈V (F3 )
i ∑ ∑ [dT (x, vj ) + 2j − (i + 1)] + [dT (x, y) − (i − 1)] y∈A
j=1
= DT (x) +
∑
(i − 1) −
y∈V (T0 )∪V (T1 )
∑
(i − 1) +
y∈A
i ∑
[2j − (i + 1)]
j=1
= DT (x) + (i − 1)[|V (T0 ) ∪ V (T1 )| − |A|] > DT (x) + (i − 1)(b − a) ≥ DT (x). (III′ ) If x ∈ V (T2 ), dT ′ (x, y) = dT (x, y) for y ∈ V (T ) \ V (F3 ). dT ′ (vj , y) = dT (vj , y) +[2j − (i + 1)] for j ∈ {1, 2, . . . , i} and y ∈ V (F3 ), and dT ′ (x, y) = dT (x, y) − (i − 1) for x ∈ A and y ∈ V (F3 ). So, DT ′ (x) = DT (x) − (i − 1)f3 for x ∈ A. DT ′ (vj ) = DT (vj ) +[2j − (i + 1)]f3 for j ∈ {1, 2, . . . , i}. Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.
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Now we calculate ξ d (T ′ ) − ξ d (T ). By (I), (II) and (I′ ),
ξ d (T ′ ) − ξ d (T ) =
∑
[εT ′ (x)DT ′ (x) − εT (x)DT (x)]
x∈V (T )
≥
∑
{εT (x)[DT ′ (x) − DT (x)]}
x∈V (F3 )
+
∑ ∑ {εT (x)[DT ′ (x) − DT (x)]} + {εT (x)[DT ′ (x) − DT (x)]} x∈B
+
k−1 ∑
x∈A
εT (vj )[DT ′ (vj ) − DT (vj )].
j=1
By (II′ ), S1 ≜
∑
{εT (x)[DT ′ (x) − DT (x)]} > 0.
(1)
x∈V (F3 )
By (I′ ), (III′ ) and the assumption that a ≤ b, S2 ≜
∑ ∑ {εT (x)[DT ′ (x) − DT (x)]} + {εT (x)[DT ′ (x) − DT (x)]} x∈B
x∈A
= k(i − 1)f3 b + k[−(i − 1)]f3 a ≥ 0. ∑k−1 Let j=1 εT (vj )[DT ′ (vj ) − DT (vj )] ≜ S3 .
(2)
When k is odd, S3 = If i ≤
∑ k−2 1
{ε v [
v − DT (vj )] + εT (vk−j )[DT ′ (vk−j ) − DT (vk−j )]}.
T ( j ) DT ′ ( j ) j=1 k−1 , by (I′ ) and (III′ ), 2
k−1
i 2 ∑ ∑ S3 = {εT (vj )[2j − (i + 1) + (i − 1)]f3 } + {εT (vj )[2(i − 1)]f3 } > 0. j=1
If i >
k−1 , 2
by (I′ ) and (III′ ), k−1
k−i−1
S3 =
∑
{εT (vj )[2j − (i + 1) + (i − 1)]f3 } +
j=1
2 ∑
{εT (vj )[2j − (i + 1) + 2(k − j) − (i + 1)]f3 }
j=k−i k−1
k−i−1
=
(3)
j=i+1
∑
[εT (vj )(2j − 2)f3 ] +
2 ∑
{εT (vj )[2k − 2(i + 1)]f3 }
j=k−i
j=1 k−1
≥
2 ∑
{εT (vj )[2k − 2(i + 1)]f3 } > 0.
(4)
j=k−i
When k is even, k−2
S3 =
2 ∑
{εT (vj )[DT ′ (vj ) − DT (vj )] + εT (vk−j )[DT ′ (vk−j ) − DT (vk−j )]}
j=1
+ εT (v k )[DT ′ (v k ) − DT (v k )]. 2
If i <
k , 2
S3 =
′
2
2
′
by (I ) and (III ), i ∑
k−2
{εT (vj )[2j − (i + 1) + (i − 1)]f3 } +
j=1
2 ∑
{εT (vj )[2(i − 1)]f3 }
j=i+1
+ εT (v k )(i − 1)f3 > 0.
(5)
2
Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.
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Fig. 3.2. An illustration of trees T and T ′ in A4 .
If i ≥
k 2
, k−i−1
S3 =
∑
{εT (vj )[2j − (i + 1) + (i − 1)]f3 }
j=1 k−2
+
2 ∑
{εT (vj )[2j − (i + 1) + 2(k − j) − (i + 1)]f3 } + εT (v k )(k − i − 1) 2
j=k−i k−2
k−i−1
=
∑
2 ∑
[εT (vj )(2j − 2)f3 ] +
{εT (vj )[2k − 2(i + 1)]f3 }
j=k−i
j=1
+ εT (v k )(k − i − 1) > 0.
(6)
2
So, by (1)–(6), ξ d (T ′ ) > ξ d (T ). This completes the proof of Lemma 3.2.
A4 : Let T be a tree of order n > 12 and P = v0 v1 · · · vk−1 vk be one of the longest paths in T with k ≥ 4. Let A = {x|x ∈ NT (v1 ) \ v2 }, B = {x|x ∈ NT (vk−1 ) \ vk−2 }, and assume that |A| = a ≤ |B| = b. Let i = min{j|dT (vj ) ≥ 3, 2 ≤ j ≤ k − 2}. Assume dT (vi ) = 3, w ∈ NT (vi ) \ {vi−1 , vi+1 } and dT (w ) ≥ 2. Let T ′ = T \ vi−1 vi + vi−1 w \ {w x|x ∈ NT (w ) \ vi } + {v1 x|x ∈ NT (w ) \ vi } \ v1 v2 + v0 v2 . Then T ′ is said to be obtained by running an A4 transformation of T . See Fig. 3.2. Lemma 3.3. Let T be a tree of order n > 12. If T ′ is a tree obtained from T by running an A4 , then we have ξ d (T ′ ) > ξ d (T ). Proof. In T , let T1 be the component of T \ vi containing vi−1 , T2 be the component of T \ {wvi , vi−1 vi } containing vi , and F3 be the forest induced by the vertices not in V (T1 ) ∪ V (T2 ) ∪ {w}. We let |V (T1 )| = t1 , |V (T2 )| = t2 and |V (F3 )| = f3 . First we consider εT (x) and εT ′ (x) for x ∈ V (T ). (I) If x ∈ A, εT ′ (x) ≥ dT ′ (x, vk ) ≥ k = εT (x). εT ′ (vj ) ≥ max{dT ′ (vj , v1 ), dT ′ (vj , vk )} = max{dT (vj , v0 ), dT (vj , vk ) + 1} ≥ εT (vj ) for j ∈ {2, 3, . . . , i − 1}. εT ′ (v1 ) ≥ {dT ′ (v1 , vk )} = k + 1 > εT (v1 ). (II) εT ′ (w ) ≥ max{dT ′ (w, x), dT ′ (w, vk )} ≥ max{dT (w, v0 ), dT (w, vk )} = εT (w ), where x ∈ V (F3 ). (III) If x ∈ V (T2 ), εT ′ (x) ≥ max{dT ′ (x, v0 ), dT ′ (x, vk )} = max{dT (x, v0 ), dT (x, vk )} = εT (x). (IV) If x ∈ V (F3 ), εT ′ (x) ≥ dT ′ (x, vk ) ≥ k + 2 ≥ εT (x) + 2. Now we consider DT (x) and DT ′ (x) for x ∈ V (T ). (I′ ) DT ′ (v1 ) = DT (v1 ) + |{v2 , . . . , vi−1 }| + 2|V (T2 )| + [1 − (i + 1)]|V (F3 )| = DT (v1 ) + (i − 2) + 2t2 − if3 . DT ′ (v0 ) = DT (v0 ) − (i − 2) − 2 − if3 = DT (v0 ) − i − if3 . DT ′ (x) = DT (x) + (i − 2) + 2t2 − if3 for x ∈ A \ v0 . DT ′ (vj ) = DT (vj ) + t2 − |{w}| − |{v0 }| + |{v1 }| + (a − 1) + [j + 1 − (i + 2 − j)]f3 = DT (vj ) + t2 − 2 + a + [2j − (i + 1)]f3 for j ∈ {2, 3, . . . , i − 1}. (II′ ) DT ′ (w ) = DT (w ) + (i + 1 − 1)f3 − |{v2 , . . . , vi−1 }| + [i − 1 − (i + 1)] = DT (w ) + if3 − (i − 2) − 2 = DT (w ) + if3 − i. (III′ ) If x ∈ V (T2 ), DT ′ (x) = DT (x) + (i + 2 − 2)f3 + (i ∑ − 2) + 2a = DT (x) + if3 + t1 + a − 1. i−1 (IV′ ) If x ∈ V (F3 ), DT ′ (x) = DT (x) + it2 + i − ia − i + j=2 [j + 1 − (i + 2 − j)] = DT (x) + it2 − ia. Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.
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Now we calculate ξ d (T ′ ) and ξ d (T ). By (I)−(IV) and (I′ )−(IV′ ),
ξ d (T ′ ) − ξ d (T ) =
∑
[εT ′ (x)DT ′ (x) − εT (x)DT (x)]
x∈V (T )
≥
∑
{εT (x)[DT ′ (x) − DT (x)]} + εT (w)[DT ′ (w) − DT (w)]
x∈V (F3 )
+ [εT ′ (v0 )DT ′ (v0 ) − εT (v0 )DT (v0 )] + [εT ′ (vk )DT ′ (vk ) − εT (vk )DT (vk )] + [εT ′ (v1 )DT ′ (v1 ) − εT (v1 )DT (v1 )] + [εT ′ (vk−1 )DT ′ (vk−1 ) − εT (vk−1 )DT (vk−1 )] ∑ ∑ + {εT (x)[DT ′ (x) − DT (x)]} + {εT (x)[DT ′ (x) − DT (x)]} x∈B\vk
+
k−2 ∑
x∈A\v0
{εT (vj )[DT ′ (vj ) − DT (vj )]}.
j=2
By (IV′ ), S1 ≜
∑
{εT (x)[DT ′ (x) − DT (x)]} =
x∈V (F3 )
∑
[εT (x)i(t2 − a)] > 0.
(7)
x∈V (F3 )
By (II′ ), S2 ≜ εT (w )[DT ′ (w ) − DT (w )] = εT (w )i(f3 − 1) ≥ 0. ′
(8)
′
By (I ) and (III ), S3 ≜ [εT ′ (v0 )DT ′ (v0 ) − εT (v0 )DT (v0 )] + [εT ′ (vk )DT ′ (vk ) − εT (vk )DT (vk )]
≥ k[−i − if3 + if3 + t1 − 1 + a] = k[t1 − i − 1 + a] ≥ 0.
(9)
By (I′ ) and (III′ ), S4 ≜ [εT ′ (v1 )DT ′ (v1 ) − εT (v1 )DT (v1 )] + [εT ′ (vk−1 )DT ′ (vk−1 ) − εT (vk−1 )DT (vk−1 )]
≥ k[(i − 2) + 2t2 − if3 + if3 + t1 − 1 + a] = k(t1 − 3 + 2t2 + a + i) > 2k.
(10)
By (I′ ) and (III′ ), S5 ≜
∑
{εT (x)[DT ′ (x) − DT (x)]} +
x∈B\vk
∑
{εT (x)[DT ′ (x) − DT (x)]}
x∈A\v0
= k[if3 + t1 − 1 + a](b − 1) + k[(i − 2) + 2t2 − if3 ](a − 1) = k{if3 [(b − 1) − (a − 1)] + (i − 2)(a − 1) + 2t2 (a − 1) + t1 (b − 1) − (b − 1) + a(b − 1)} ≥ 0. ∑k−2
(11)
′ Let S6 ≜ j=2 εT (vj )[DT (vj ) − DT (vj )]. If k is even, k−2
S6 =
2 ∑
{εT (vj )[DT ′ (vj ) − DT (vj )] + εT (vk−j )[DT ′ (vk−j ) − DT (vk−j )]}
j=2
+ εT (v k )[DT ′ (v k ) − DT (v k )]. 2
If i ≤
k , 2
S6 ≥
′
2
2
′
by (I ) and (III ), i−1 ∑ {εT (vj ){t2 + a − 2 + [2j − (i + 1)]f3 + if3 + t1 − 1 + a}} j=2 k−2 2 ∑ + [2εT (vj )(if3 + t1 − 1 + a)] + εT (v k )(if3 + t1 − 1 + a) > 0.
(12)
2
j=i
Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.
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If i − 1 ≥ S6 ≥
k , 2
)
–
7
by (I′ ) and (III′ ),
k−i ∑ {εT (vj ){t2 + a − 2 + [2j − (i + 1)]f3 + if3 + t1 − 1 + a}} j=2 k
−1 2 ∑
+
{εT (vj ){(t2 − 2 + a) × 2 + [2j − (i + 1) + 2(k − j) − (i + 1)]f3 }}
j=k−i+1
+ εT (v k ){t2 − 2 + a + [k − (i + 1)]}f3 > 0.
(13)
2
∑ k−1
2 ′ ′ If k is odd, S6 = j=2 {εT (vj )[DT (vj ) − DT (vj )] + εT (vk−j )[DT (vk−j ) − DT (vk−j )]}. k−1 ′ ′ If i − 1 < 2 , by (I ) and (III ),
S6 ≥
i−1 ∑ {εT (vj ){t2 + a − 2 + [2j − (i + 1)]f3 + if3 + t1 − 1 + a}}
(14)
j=2 k−1
+
2 ∑
[2εT (vj )(if3 + t1 − 1 + a)] > 0.
(15)
j=i
If i − 1 ≥ S6 ≥
k−1 , 2
by (I′ ) and (III′ ),
k−i ∑ {εT (vj ){t2 + a − 2 + [2j − (i + 1)]f3 + if3 + t1 − 1 + a}} j=2 k−1
+
2 ∑
{εT (vj ){2t2 − 4 + 2a + [2j − (i + 1) + 2(k − j) − (i + 1)]f3 }} > 0.
(16)
j=k−i+1
By (7)–(16), ξ d (T ′ ) > ξ d (T ). This completes the proof of Lemma 3.3. Proof of Theorem 3.1. Assume that T ∈ Tn,γ has the maximal EDS. Let v0 v1 · · · vk−1 vk be one of the longest paths in T . Then k ≥ 4, for otherwise γ (T ) ≤ 2. If {j|dT (vj ) ≥ 3, 2 ≤ j ≤ k−2} = ∅, T ∼ = Pk−2 (p, q) for some integers p, q, and k ≤ 3γ (T ). By Lemmas 2.1 and 2.6, when T ∼ = Pl (⌊ n2−l ⌋, ⌈ n2−l ⌉) , where l = 3γ (T )−2, T has the maximal EDS. Now, we assume that {j|dT (vj ) ≥ 3, 2 ≤ j ≤ k − 2} ̸ = ∅. Let C = {v1 , . . . , vk−1 }, A = {x|x ∈ NT (v1 ) \ v2 }, B = {x|x ∈ NT (vk−1 ) \ vk−2 }, and assume that |A| = a ≤ |B| = b. Let i = min{j|dT (vj ) ≥ 3, 2 ≤ j ≤ k − 2}. Let S be a minimum dominating set of T such that the leaves of T are not in S and |S ∩ {vi−1 , vi , vi+1 }| is as large as possible. We consider two cases. Case 1. {vi−1 , vi , vi+1 } ∩ S ̸ = ∅. Let T1 = T \ {vi x|x ∈ NT (vi ) \ {vi−1 , vi+1 }} + {v1 x|x ∈ NT (vi ) \ {vi−1 , vi+1 }}. Then S is also a dominating set of T1 . So γ (T1 ) ≤ γ (T ). By Lemma 2.2, ξ d (T1 ) > ξ d (T ). Case 2. {vi−1 , vi , vi+1 } ∩ S = ∅. In this case, there exists w ∈ NT (vi ) \ {vi−1 , vi+1 } such that w ∈ S and dT (w ) ≥ 2 since the leaves of T are not in S. We consider two subcases. Subcase 2.1. dT (vi ) ≥ 4. Let T1 = T \{vi y|y ∈ NT (vi ) \{vi−1 , vi+1 , w}}+{v1 y|y ∈ NT (vi ) \{vi−1 , vi+1 , w}}. S is a dominating set of T1 , so γ (T1 ) ≤ γ (T ). By Lemma 3.2, ξ d (T1 ) > ξ d (T ). Subcase 2.2. dT (vi ) = 3. Since S is a minimum dominate set and by the assumptions that v1 ∈ S and {vi−1 , vi , vi+1 } ∩ S = ∅, we can get i ≡ 0 (mod 3), and for all j ∈ {1, 2, . . . , i}, vj ∈ S if j ≡ 1 (mod 3). Let T1 = T \ vi−1 vi + wvi−1 \ {w x|x ∈ NT (w ) \ vi } + {v1 x|x ∈ NT (w ) \ vi } \ v1 v2 + v0 v2 and S ′ = S \ {vj |1 < j < i, j ≡ 1( mod 3)} + {vj |1 < j < i, j ≡ 0( mod 3)}. Then |S ′ | = |S | and S ′ is a dominating set of T1 , so γ (T1 ) ≤ γ (T ). By Lemma 3.3, ξ d (T1 ) > ξ d (T ). If T1 ≇ Pc (p′ , q′ ) for any integers c , p′ , q′ , we continue the transformation of A2 , A3 or A4 for T1 to get a tree T2 such that d ξ (T2 ) > ξ d (T1 ) and γ (T2 ) ≤ γ (T1 ). Because Pn maximizes the EDS among n-vertex trees, the procedure can stop at some step, that is, we will get a series of trees T , T1 , . . . , Tl such that γ (Ti+1 ) ≤ γ (Ti ) ≤ γ (T ), ξ d (Ti+1 ) > ξ d (Ti ) > ξ d (T ) for 1 ≤ i ≤ l − 1 and Tl ∼ = Pc (p′ , q′ ) for some integers c , p′ , q′ . Noting that diam(Tl ) ≤ 3γ (Tl ) ≤ 3γ (T ), if diam(Tl ) < 3γ (T ), we can run some times of A1 transformation of Tl until we get a tree Tl∗ such that diam(Tl∗ ) = 3γ (T ). By Lemma 2.1, ξ d (Tl∗ ) > ξ d (Tl ). n−f n−f Let Tl∗ ∼ = P3γ (T )−2 (r , s) for some integers r and s. By Lemma 2.6, ξ d (Tl∗ ) ≤ ξ d (Pf (⌊ 2 ⌋, ⌈ 2 ⌉))(f = 3γ (T ) − 2). Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.
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So ξ d (Pf (⌊ 2 ⌋, ⌈ 2 ⌉)) > ξ d (T ) and γ (Pf (⌊ 2 ⌋, ⌈ 2 ⌉)) = γ (T ) (f = 3γ (T ) − 2), a contradiction to the assumption that T has the maximal EDS among Tn,γ (4 ≤ γ < ⌈ 3n ⌉). This completes the proof of Theorem 3.1. n−f
n−f
n−f
n−f
Acknowledgments The authors would like to express their thanks to the referees for their valuable corrections and suggestions of the manuscript that greatly improve the format and correctness of it. The first author was supported by National Natural Science Foundation of China (11271365) and partially supported by National Natural Science Foundation of China (11571258). References [1] 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. [2] X.Y. Geng, S.C. Li, M. Zhang, Extremal values on the eccentric distance sum of trees, Discrete Appl. Math. 161 (2013) 2427–2439. [3] 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. [4] H.B. Hua, K.X. Xu, S. Wen, 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. [5] H.B. Hua, S.G. Zhang, K.X. Xu, Further results on the eccentric distance sum, Discrete Appl. Math. 160 (2012) 170–180. [6] A. Ilić, I. Gutman, Eccentric connectivity index of chemical trees, MATCH Commun. Math. Comput. Chem. 65 (2011) 731–744. [7] A. Ilić, G.H. Yu, L.H. Feng, On the eccentric distance sum of graphs, J. Math. Anal. Appl. 381 (2011) 590–600. [8] S. Li, Sharp bounds on the eccentric distance sum of graphs, in: I. Gutman, B. Furtula, K.C. Das, E. Milovanović, I. Milovanović (Eds.), Bounds in Chemical Graph Theory—Mainstreams, Univ. Kragujevac, Kragujevac, 2017, pp. 207–237. [9] 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. [10] L.Y. Miao, Q.Q. Cao, N. C, S.Y. Pang, On the extremal values of the eccentric distance sum of trees, Discrete Appl. Math. 186 (2015) 199–206. [11] M.J. Morgan, S. Mukwembi, H.C. Swart, A lower bound on the eccentric connectivity index of a graph, Discrete Appl. Math. 160 (2012) 248–258. [12] O. Ore, Theory of graphs, Amer. Math. Soc. Colloq. Publ. 38 (1962). [13] 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. [14] K. Xu, Ch. Das, A.D. Maden, On a novel eccentricity–based invariant of a graph, Acta Math. Sin. (Engl. Ser.) 32 (2016) 1477–1493. [15] K. Xu, K.C. Das, H. Liu, Some extremal results on the connective eccentricity index of graphs, J. Math. Anal. Appl. 433 (2016) 803–817. [16] K.X. Xu, L.H. Feng, Extremal energies of trees with a given domination number, Linear Algebra Appl. 435 (2011) 2382–2393. [17] G.H. Yu, L.H. Feng, On the connective eccentricity index of graphs, MATCH Commun. Math. Comput. Chem. 69 (2013) 611–628. [18] 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. [19] G. Yu, H. Qu, L. Tang, L. Feng, On the connective eccentricity index of trees and unicyclic graphs with given diameter, J. Math. Anal. Appl. 420 (2014) 1776–1786. [20] J. Zhang, B. Zhou, Z. Liu, On the minimal eccentric connectivity indices of graphs, Discrete Math. 312 (2012) 819–829.
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On the extremal values of the eccentric distance sum of trees with a given domination number Lianying Miao *, Shiyou Pang, Fang Liu, Eryan Wang, Xiaoqing Guo School of Science, China University of Mining and Technology, Xuzhou, Jiangsu, 221008, PR China
article
info
Article history: Received 13 March 2016 Received in revised form 24 April 2017 Accepted 26 April 2017 Available online xxxx Keywords: Tree Eccentric distance sum Domination number
a b s t r a c t Let G be a∑ simple connected graph. The eccentric distance sum (EDS) of G is defined as d ξ∑ (G) = 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 extremal tree among n-vertex trees with domination number γ satisfying 4 ≤ γ < ⌈ 3n ⌉ having the maximal EDS is characterized. This proves Conjecture 4.2 posed in Miao et al. (2015). © 2017 Elsevier B.V. All rights reserved.
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 for short) to denote the maximum degree and minimum degree of G, respectively. For u, v ∈ V , the distance dG (u, v ) is defined as the length of a shortest path between u and v in G. DG (v ) denotes the sum of all distances from v . The eccentricity εG (v ) of a vertex v is the maximum distance from v to any other vertex. The diam(G) of a graph G is the maximum eccentricity of any vertex in G. 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}. For E ′ ⊆ E, G \ E ′ denotes the graph obtained from G by deleting the edges in E ′ . If E ′ = {e}, we just write G \ e for G − {e}. 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 Pn be a path on n vertices. We use l(P) to denote the length of a path P. For a real number x, we use ⌊x⌋ to denote the greatest integer no greater than x and use ⌈x⌉ to denote the least integer no less than x. 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. 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 [12] obtained that γ (G) ≤ 2n . Some results on the domination number can be found in [1,13,16]. The graph invariant-eccentric distance sum (EDS) was introduced by Gupta, Singh and Madan [3], which was defined as
ξ d (G) =
∑
εG (v )DG (v ).
v∈V
*
Corresponding author. E-mail address: [email protected] (L. Miao).
http://dx.doi.org/10.1016/j.dam.2017.04.032 0166-218X/© 2017 Elsevier B.V. All rights reserved.
Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.
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Fig. 2.1. An illustration of trees T and T ′ in A2 .
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 [18] characterized the trees with the minimal EDS among the n-vertex trees of a given diameter. Li, Zhang, Yu, Feng [9] 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 trees with the second, third and fourth minimal eccentric distance sum among the n-vertex trees of a given diameter. Geng, Li and Zhang [2] and Miao [10] 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 and 3n . They also identify the trees with the minimal and the maximal EDS among the n-vertex trees with k leaves. In [10] the trees having the maximal EDS among n-vertex trees with maximum degree ∆ 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, respectively. Other results about the EDS of graphs can also be found in [4,5,7,8]. For other eccentricity-based graph invariants one may refer to [6,11,14,15,17,19,20]. In this paper, we continue to study the eccentric distance sum of trees. The extremal tree among n-vertex trees with domination number γ satisfying 4 ≤ γ < ⌈ 3n ⌉ having the maximal EDS is characterized. This proves Conjecture 4.2 posed in [10]. 2. Preliminaries The following graph transformations A1 and A2 were posed in [5] and [2] respectively. A1 : Let T be a tree of order n > 3 and e = uv be a nonpendant 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 ([5]). Let T be a tree of order n > 3 and e = uv be a nonpendant 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 : Suppose that P = v0 v1 · · · vi · · · vk is one of the longest paths contained in an n-vertex tree T with dT (v1 ) ≤ dT (vk−1 ) and i = min{j|dT (vj ) ≥ 3, j = 2, 3, . . ., k − 2}. Let T ′ = T − {vi u|u ∈ NT (vi ) \ {vi−1 , vi+1 }} + {v1 u|u ∈ NT (vi ) \ {vi−1 , vi+1 }}. Then T ′ is said to be obtained by running an A2 transformation of T . See Fig. 2.1. Lemma 2.2 ([2]). Let T be an n-vertex tree. If T ′ is a tree obtained from T by running an A2 , then we have ξ d (T ′ ) > ξ d (T ). Let Tn,γ be the set of all n-vertex trees with domination number γ . 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. Lemma 2.3 ([2]). Among all the trees from Tn,⌈ n ⌉ with n > 4, the tree Pn has the maximal EDS. 3
4 Lemma 2.4 ([2]). Among Tn,2 with n ≥ 4, P4 (⌊ n− ⌋, ⌈ n−2 4 ⌉) maximizes the EDS. 2 7 Lemma 2.5 ([10]). Among Tn,3 with n ≥ 10, P7 (⌊ n− ⌋, ⌈ n−2 7 ⌉) maximizes the EDS. 2
Lemma 2.6 ([2]). ξ d (Pl (1, n − l − 1)) < ξ d (Pl (2, n − l − 2)) < · · · < ξ d (Pl (⌊ n2−l ⌋, ⌈ n2−l ⌉)).
Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.
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Fig. 3.1. An illustration of trees T and T ′ in A3 .
3. The extremal EDS of n-vertex tree with domination number at most ⌈ n3 ⌉ By Lemmas 2.3–2.5, we assume that 4 ≤ γ < ⌈ 3n ⌉. Let T ∈ Tn,γ with 4 ≤ γ < ⌈ 3n ⌉. By 4 < ⌈ 3n ⌉, n ≥ 13. From now on, we assume that n ≥ 13. We have the following theorem. Theorem 3.1. Among Tn,γ (4 ≤ γ < ⌈ 3n ⌉), Pl (⌊ n2−l ⌋, ⌈ n2−l ⌉) maximizes the EDS, where l = 3γ − 2. Before the proof of Theorem 3.1, we introduce two graph transformations A3 and A4 as follows. A3 : Let T be an n-vertex tree of order n > 12 and P = v0 v1 · · · vk−1 vk be one of the longest paths in T with k ≥ 4. Let A = {x|x ∈ NT (v1 ) \ v2 }, B = {x|x ∈ NT (vk−1 ) \ vk−2 }, and assume that |A| = a ≤ |B| = b. Let i = min{j|dT (vj ) ≥ 3, 2 ≤ j ≤ k − 2}. Assume dT (vi ) ≥ 4 and w ∈ NT (vi ) \{vi−1 , vi+1 }. Let T ′ = T \{vi y|y ∈ NT (vi ) \{vi−1 , vi+1 , w}}+{v1 y|y ∈ NT (vi ) \{vi−1 , vi+1 , w}}. Then T ′ is said to be obtained by running an A3 transformation of T . See Fig. 3.1. Lemma 3.2. Let T be an n-vertex tree. If T ′ is a tree obtained from T by running an A3 , then we have ξ d (T ′ ) > ξ d (T ). Proof. In T , let T0 be the component of T \ vi containing w , T1 be the component of T \ vi containing vi+1 , T2 be the tree induced by A ∪ {v1 , v2 , . . . , vi } and F3 be the forest induced by the vertices not in V (T0 ) ∪ V (T1 ) ∪ V (T2 ). We let |V (F3 )| = f3 . First we consider εT (x) and εT ′ (x) for x ∈ V (T ). (I) If x ∈ V (T ) \ V (F3 ), εT ′ (x) ≥ max{dT ′ (x, v0 ), dT ′ (x, vk )} = max{dT (x, v0 ), dT (x, vk )} = εT (x). (II) If x ∈ V (F3 ), εT ′ (x) ≥ dT (v0 , vk ) = εT (v0 ) ≥ εT (x). Now we consider DT (x) and DT ′ (x) for x ∈ V (T ). (I′ ) If x ∈ V (T0 ) ∪ V (T1 ), dT ′ (x, y) = dT (x, y) for y ∈ V (T ) \ V (F3 ) and dT ′ (x, y) = dT (x, y) + i − 1 ≥ dT (x, y) for y ∈ V (F3 ). So DT ′ (x) = DT (x) + (i − 1)f3 > DT (x). (II′ ) If x ∈ V (F3 ), dT ′ (x, y) = dT (x, y) + (i − 1) for y ∈ V (T0 ) ∪ V (T1 ), dT ′ (x, y) = dT (x, y) for y ∈ V (F3 ), dT ′ (x, vj ) = dT (x, vj ) + 2j − (i + 1) for j ∈ {1, 2, . . . , i} and dT ′ (x, y) = dT (x, y) − (i − 1) for y ∈ A. So, by the fact that A ⊂ V (T2 ), B ⊂ V (T1 ) and a ≤ b,
∑
DT ′ (x) =
[dT (x, y) + (i − 1)] +
y∈V (T0 )∪V (T1 )
+
∑
dT (x, y)
y∈V (F3 )
i ∑ ∑ [dT (x, vj ) + 2j − (i + 1)] + [dT (x, y) − (i − 1)] y∈A
j=1
= DT (x) +
∑
(i − 1) −
y∈V (T0 )∪V (T1 )
∑
(i − 1) +
y∈A
i ∑
[2j − (i + 1)]
j=1
= DT (x) + (i − 1)[|V (T0 ) ∪ V (T1 )| − |A|] > DT (x) + (i − 1)(b − a) ≥ DT (x). (III′ ) If x ∈ V (T2 ), dT ′ (x, y) = dT (x, y) for y ∈ V (T ) \ V (F3 ). dT ′ (vj , y) = dT (vj , y) +[2j − (i + 1)] for j ∈ {1, 2, . . . , i} and y ∈ V (F3 ), and dT ′ (x, y) = dT (x, y) − (i − 1) for x ∈ A and y ∈ V (F3 ). So, DT ′ (x) = DT (x) − (i − 1)f3 for x ∈ A. DT ′ (vj ) = DT (vj ) +[2j − (i + 1)]f3 for j ∈ {1, 2, . . . , i}. Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.
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Now we calculate ξ d (T ′ ) − ξ d (T ). By (I), (II) and (I′ ),
ξ d (T ′ ) − ξ d (T ) =
∑
[εT ′ (x)DT ′ (x) − εT (x)DT (x)]
x∈V (T )
≥
∑
{εT (x)[DT ′ (x) − DT (x)]}
x∈V (F3 )
+
∑ ∑ {εT (x)[DT ′ (x) − DT (x)]} + {εT (x)[DT ′ (x) − DT (x)]} x∈B
+
k−1 ∑
x∈A
εT (vj )[DT ′ (vj ) − DT (vj )].
j=1
By (II′ ), S1 ≜
∑
{εT (x)[DT ′ (x) − DT (x)]} > 0.
(1)
x∈V (F3 )
By (I′ ), (III′ ) and the assumption that a ≤ b, S2 ≜
∑ ∑ {εT (x)[DT ′ (x) − DT (x)]} + {εT (x)[DT ′ (x) − DT (x)]} x∈B
x∈A
= k(i − 1)f3 b + k[−(i − 1)]f3 a ≥ 0. ∑k−1 Let j=1 εT (vj )[DT ′ (vj ) − DT (vj )] ≜ S3 .
(2)
When k is odd, S3 = If i ≤
∑ k−2 1
{ε v [
v − DT (vj )] + εT (vk−j )[DT ′ (vk−j ) − DT (vk−j )]}.
T ( j ) DT ′ ( j ) j=1 k−1 , by (I′ ) and (III′ ), 2
k−1
i 2 ∑ ∑ S3 = {εT (vj )[2j − (i + 1) + (i − 1)]f3 } + {εT (vj )[2(i − 1)]f3 } > 0. j=1
If i >
k−1 , 2
by (I′ ) and (III′ ), k−1
k−i−1
S3 =
∑
{εT (vj )[2j − (i + 1) + (i − 1)]f3 } +
j=1
2 ∑
{εT (vj )[2j − (i + 1) + 2(k − j) − (i + 1)]f3 }
j=k−i k−1
k−i−1
=
(3)
j=i+1
∑
[εT (vj )(2j − 2)f3 ] +
2 ∑
{εT (vj )[2k − 2(i + 1)]f3 }
j=k−i
j=1 k−1
≥
2 ∑
{εT (vj )[2k − 2(i + 1)]f3 } > 0.
(4)
j=k−i
When k is even, k−2
S3 =
2 ∑
{εT (vj )[DT ′ (vj ) − DT (vj )] + εT (vk−j )[DT ′ (vk−j ) − DT (vk−j )]}
j=1
+ εT (v k )[DT ′ (v k ) − DT (v k )]. 2
If i <
k , 2
S3 =
′
2
2
′
by (I ) and (III ), i ∑
k−2
{εT (vj )[2j − (i + 1) + (i − 1)]f3 } +
j=1
2 ∑
{εT (vj )[2(i − 1)]f3 }
j=i+1
+ εT (v k )(i − 1)f3 > 0.
(5)
2
Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.
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)
–
5
Fig. 3.2. An illustration of trees T and T ′ in A4 .
If i ≥
k 2
, k−i−1
S3 =
∑
{εT (vj )[2j − (i + 1) + (i − 1)]f3 }
j=1 k−2
+
2 ∑
{εT (vj )[2j − (i + 1) + 2(k − j) − (i + 1)]f3 } + εT (v k )(k − i − 1) 2
j=k−i k−2
k−i−1
=
∑
2 ∑
[εT (vj )(2j − 2)f3 ] +
{εT (vj )[2k − 2(i + 1)]f3 }
j=k−i
j=1
+ εT (v k )(k − i − 1) > 0.
(6)
2
So, by (1)–(6), ξ d (T ′ ) > ξ d (T ). This completes the proof of Lemma 3.2.
A4 : Let T be a tree of order n > 12 and P = v0 v1 · · · vk−1 vk be one of the longest paths in T with k ≥ 4. Let A = {x|x ∈ NT (v1 ) \ v2 }, B = {x|x ∈ NT (vk−1 ) \ vk−2 }, and assume that |A| = a ≤ |B| = b. Let i = min{j|dT (vj ) ≥ 3, 2 ≤ j ≤ k − 2}. Assume dT (vi ) = 3, w ∈ NT (vi ) \ {vi−1 , vi+1 } and dT (w ) ≥ 2. Let T ′ = T \ vi−1 vi + vi−1 w \ {w x|x ∈ NT (w ) \ vi } + {v1 x|x ∈ NT (w ) \ vi } \ v1 v2 + v0 v2 . Then T ′ is said to be obtained by running an A4 transformation of T . See Fig. 3.2. Lemma 3.3. Let T be a tree of order n > 12. If T ′ is a tree obtained from T by running an A4 , then we have ξ d (T ′ ) > ξ d (T ). Proof. In T , let T1 be the component of T \ vi containing vi−1 , T2 be the component of T \ {wvi , vi−1 vi } containing vi , and F3 be the forest induced by the vertices not in V (T1 ) ∪ V (T2 ) ∪ {w}. We let |V (T1 )| = t1 , |V (T2 )| = t2 and |V (F3 )| = f3 . First we consider εT (x) and εT ′ (x) for x ∈ V (T ). (I) If x ∈ A, εT ′ (x) ≥ dT ′ (x, vk ) ≥ k = εT (x). εT ′ (vj ) ≥ max{dT ′ (vj , v1 ), dT ′ (vj , vk )} = max{dT (vj , v0 ), dT (vj , vk ) + 1} ≥ εT (vj ) for j ∈ {2, 3, . . . , i − 1}. εT ′ (v1 ) ≥ {dT ′ (v1 , vk )} = k + 1 > εT (v1 ). (II) εT ′ (w ) ≥ max{dT ′ (w, x), dT ′ (w, vk )} ≥ max{dT (w, v0 ), dT (w, vk )} = εT (w ), where x ∈ V (F3 ). (III) If x ∈ V (T2 ), εT ′ (x) ≥ max{dT ′ (x, v0 ), dT ′ (x, vk )} = max{dT (x, v0 ), dT (x, vk )} = εT (x). (IV) If x ∈ V (F3 ), εT ′ (x) ≥ dT ′ (x, vk ) ≥ k + 2 ≥ εT (x) + 2. Now we consider DT (x) and DT ′ (x) for x ∈ V (T ). (I′ ) DT ′ (v1 ) = DT (v1 ) + |{v2 , . . . , vi−1 }| + 2|V (T2 )| + [1 − (i + 1)]|V (F3 )| = DT (v1 ) + (i − 2) + 2t2 − if3 . DT ′ (v0 ) = DT (v0 ) − (i − 2) − 2 − if3 = DT (v0 ) − i − if3 . DT ′ (x) = DT (x) + (i − 2) + 2t2 − if3 for x ∈ A \ v0 . DT ′ (vj ) = DT (vj ) + t2 − |{w}| − |{v0 }| + |{v1 }| + (a − 1) + [j + 1 − (i + 2 − j)]f3 = DT (vj ) + t2 − 2 + a + [2j − (i + 1)]f3 for j ∈ {2, 3, . . . , i − 1}. (II′ ) DT ′ (w ) = DT (w ) + (i + 1 − 1)f3 − |{v2 , . . . , vi−1 }| + [i − 1 − (i + 1)] = DT (w ) + if3 − (i − 2) − 2 = DT (w ) + if3 − i. (III′ ) If x ∈ V (T2 ), DT ′ (x) = DT (x) + (i + 2 − 2)f3 + (i ∑ − 2) + 2a = DT (x) + if3 + t1 + a − 1. i−1 (IV′ ) If x ∈ V (F3 ), DT ′ (x) = DT (x) + it2 + i − ia − i + j=2 [j + 1 − (i + 2 − j)] = DT (x) + it2 − ia. Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.
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)
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Now we calculate ξ d (T ′ ) and ξ d (T ). By (I)−(IV) and (I′ )−(IV′ ),
ξ d (T ′ ) − ξ d (T ) =
∑
[εT ′ (x)DT ′ (x) − εT (x)DT (x)]
x∈V (T )
≥
∑
{εT (x)[DT ′ (x) − DT (x)]} + εT (w)[DT ′ (w) − DT (w)]
x∈V (F3 )
+ [εT ′ (v0 )DT ′ (v0 ) − εT (v0 )DT (v0 )] + [εT ′ (vk )DT ′ (vk ) − εT (vk )DT (vk )] + [εT ′ (v1 )DT ′ (v1 ) − εT (v1 )DT (v1 )] + [εT ′ (vk−1 )DT ′ (vk−1 ) − εT (vk−1 )DT (vk−1 )] ∑ ∑ + {εT (x)[DT ′ (x) − DT (x)]} + {εT (x)[DT ′ (x) − DT (x)]} x∈B\vk
+
k−2 ∑
x∈A\v0
{εT (vj )[DT ′ (vj ) − DT (vj )]}.
j=2
By (IV′ ), S1 ≜
∑
{εT (x)[DT ′ (x) − DT (x)]} =
x∈V (F3 )
∑
[εT (x)i(t2 − a)] > 0.
(7)
x∈V (F3 )
By (II′ ), S2 ≜ εT (w )[DT ′ (w ) − DT (w )] = εT (w )i(f3 − 1) ≥ 0. ′
(8)
′
By (I ) and (III ), S3 ≜ [εT ′ (v0 )DT ′ (v0 ) − εT (v0 )DT (v0 )] + [εT ′ (vk )DT ′ (vk ) − εT (vk )DT (vk )]
≥ k[−i − if3 + if3 + t1 − 1 + a] = k[t1 − i − 1 + a] ≥ 0.
(9)
By (I′ ) and (III′ ), S4 ≜ [εT ′ (v1 )DT ′ (v1 ) − εT (v1 )DT (v1 )] + [εT ′ (vk−1 )DT ′ (vk−1 ) − εT (vk−1 )DT (vk−1 )]
≥ k[(i − 2) + 2t2 − if3 + if3 + t1 − 1 + a] = k(t1 − 3 + 2t2 + a + i) > 2k.
(10)
By (I′ ) and (III′ ), S5 ≜
∑
{εT (x)[DT ′ (x) − DT (x)]} +
x∈B\vk
∑
{εT (x)[DT ′ (x) − DT (x)]}
x∈A\v0
= k[if3 + t1 − 1 + a](b − 1) + k[(i − 2) + 2t2 − if3 ](a − 1) = k{if3 [(b − 1) − (a − 1)] + (i − 2)(a − 1) + 2t2 (a − 1) + t1 (b − 1) − (b − 1) + a(b − 1)} ≥ 0. ∑k−2
(11)
′ Let S6 ≜ j=2 εT (vj )[DT (vj ) − DT (vj )]. If k is even, k−2
S6 =
2 ∑
{εT (vj )[DT ′ (vj ) − DT (vj )] + εT (vk−j )[DT ′ (vk−j ) − DT (vk−j )]}
j=2
+ εT (v k )[DT ′ (v k ) − DT (v k )]. 2
If i ≤
k , 2
S6 ≥
′
2
2
′
by (I ) and (III ), i−1 ∑ {εT (vj ){t2 + a − 2 + [2j − (i + 1)]f3 + if3 + t1 − 1 + a}} j=2 k−2 2 ∑ + [2εT (vj )(if3 + t1 − 1 + a)] + εT (v k )(if3 + t1 − 1 + a) > 0.
(12)
2
j=i
Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.
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If i − 1 ≥ S6 ≥
k , 2
)
–
7
by (I′ ) and (III′ ),
k−i ∑ {εT (vj ){t2 + a − 2 + [2j − (i + 1)]f3 + if3 + t1 − 1 + a}} j=2 k
−1 2 ∑
+
{εT (vj ){(t2 − 2 + a) × 2 + [2j − (i + 1) + 2(k − j) − (i + 1)]f3 }}
j=k−i+1
+ εT (v k ){t2 − 2 + a + [k − (i + 1)]}f3 > 0.
(13)
2
∑ k−1
2 ′ ′ If k is odd, S6 = j=2 {εT (vj )[DT (vj ) − DT (vj )] + εT (vk−j )[DT (vk−j ) − DT (vk−j )]}. k−1 ′ ′ If i − 1 < 2 , by (I ) and (III ),
S6 ≥
i−1 ∑ {εT (vj ){t2 + a − 2 + [2j − (i + 1)]f3 + if3 + t1 − 1 + a}}
(14)
j=2 k−1
+
2 ∑
[2εT (vj )(if3 + t1 − 1 + a)] > 0.
(15)
j=i
If i − 1 ≥ S6 ≥
k−1 , 2
by (I′ ) and (III′ ),
k−i ∑ {εT (vj ){t2 + a − 2 + [2j − (i + 1)]f3 + if3 + t1 − 1 + a}} j=2 k−1
+
2 ∑
{εT (vj ){2t2 − 4 + 2a + [2j − (i + 1) + 2(k − j) − (i + 1)]f3 }} > 0.
(16)
j=k−i+1
By (7)–(16), ξ d (T ′ ) > ξ d (T ). This completes the proof of Lemma 3.3. Proof of Theorem 3.1. Assume that T ∈ Tn,γ has the maximal EDS. Let v0 v1 · · · vk−1 vk be one of the longest paths in T . Then k ≥ 4, for otherwise γ (T ) ≤ 2. If {j|dT (vj ) ≥ 3, 2 ≤ j ≤ k−2} = ∅, T ∼ = Pk−2 (p, q) for some integers p, q, and k ≤ 3γ (T ). By Lemmas 2.1 and 2.6, when T ∼ = Pl (⌊ n2−l ⌋, ⌈ n2−l ⌉) , where l = 3γ (T )−2, T has the maximal EDS. Now, we assume that {j|dT (vj ) ≥ 3, 2 ≤ j ≤ k − 2} ̸ = ∅. Let C = {v1 , . . . , vk−1 }, A = {x|x ∈ NT (v1 ) \ v2 }, B = {x|x ∈ NT (vk−1 ) \ vk−2 }, and assume that |A| = a ≤ |B| = b. Let i = min{j|dT (vj ) ≥ 3, 2 ≤ j ≤ k − 2}. Let S be a minimum dominating set of T such that the leaves of T are not in S and |S ∩ {vi−1 , vi , vi+1 }| is as large as possible. We consider two cases. Case 1. {vi−1 , vi , vi+1 } ∩ S ̸ = ∅. Let T1 = T \ {vi x|x ∈ NT (vi ) \ {vi−1 , vi+1 }} + {v1 x|x ∈ NT (vi ) \ {vi−1 , vi+1 }}. Then S is also a dominating set of T1 . So γ (T1 ) ≤ γ (T ). By Lemma 2.2, ξ d (T1 ) > ξ d (T ). Case 2. {vi−1 , vi , vi+1 } ∩ S = ∅. In this case, there exists w ∈ NT (vi ) \ {vi−1 , vi+1 } such that w ∈ S and dT (w ) ≥ 2 since the leaves of T are not in S. We consider two subcases. Subcase 2.1. dT (vi ) ≥ 4. Let T1 = T \{vi y|y ∈ NT (vi ) \{vi−1 , vi+1 , w}}+{v1 y|y ∈ NT (vi ) \{vi−1 , vi+1 , w}}. S is a dominating set of T1 , so γ (T1 ) ≤ γ (T ). By Lemma 3.2, ξ d (T1 ) > ξ d (T ). Subcase 2.2. dT (vi ) = 3. Since S is a minimum dominate set and by the assumptions that v1 ∈ S and {vi−1 , vi , vi+1 } ∩ S = ∅, we can get i ≡ 0 (mod 3), and for all j ∈ {1, 2, . . . , i}, vj ∈ S if j ≡ 1 (mod 3). Let T1 = T \ vi−1 vi + wvi−1 \ {w x|x ∈ NT (w ) \ vi } + {v1 x|x ∈ NT (w ) \ vi } \ v1 v2 + v0 v2 and S ′ = S \ {vj |1 < j < i, j ≡ 1( mod 3)} + {vj |1 < j < i, j ≡ 0( mod 3)}. Then |S ′ | = |S | and S ′ is a dominating set of T1 , so γ (T1 ) ≤ γ (T ). By Lemma 3.3, ξ d (T1 ) > ξ d (T ). If T1 ≇ Pc (p′ , q′ ) for any integers c , p′ , q′ , we continue the transformation of A2 , A3 or A4 for T1 to get a tree T2 such that d ξ (T2 ) > ξ d (T1 ) and γ (T2 ) ≤ γ (T1 ). Because Pn maximizes the EDS among n-vertex trees, the procedure can stop at some step, that is, we will get a series of trees T , T1 , . . . , Tl such that γ (Ti+1 ) ≤ γ (Ti ) ≤ γ (T ), ξ d (Ti+1 ) > ξ d (Ti ) > ξ d (T ) for 1 ≤ i ≤ l − 1 and Tl ∼ = Pc (p′ , q′ ) for some integers c , p′ , q′ . Noting that diam(Tl ) ≤ 3γ (Tl ) ≤ 3γ (T ), if diam(Tl ) < 3γ (T ), we can run some times of A1 transformation of Tl until we get a tree Tl∗ such that diam(Tl∗ ) = 3γ (T ). By Lemma 2.1, ξ d (Tl∗ ) > ξ d (Tl ). n−f n−f Let Tl∗ ∼ = P3γ (T )−2 (r , s) for some integers r and s. By Lemma 2.6, ξ d (Tl∗ ) ≤ ξ d (Pf (⌊ 2 ⌋, ⌈ 2 ⌉))(f = 3γ (T ) − 2). Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.
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So ξ d (Pf (⌊ 2 ⌋, ⌈ 2 ⌉)) > ξ d (T ) and γ (Pf (⌊ 2 ⌋, ⌈ 2 ⌉)) = γ (T ) (f = 3γ (T ) − 2), a contradiction to the assumption that T has the maximal EDS among Tn,γ (4 ≤ γ < ⌈ 3n ⌉). This completes the proof of Theorem 3.1. n−f
n−f
n−f
n−f
Acknowledgments The authors would like to express their thanks to the referees for their valuable corrections and suggestions of the manuscript that greatly improve the format and correctness of it. The first author was supported by National Natural Science Foundation of China (11271365) and partially supported by National Natural Science Foundation of China (11571258). References [1] 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. [2] X.Y. Geng, S.C. Li, M. Zhang, Extremal values on the eccentric distance sum of trees, Discrete Appl. Math. 161 (2013) 2427–2439. [3] 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. [4] H.B. Hua, K.X. Xu, S. Wen, 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. [5] H.B. Hua, S.G. Zhang, K.X. Xu, Further results on the eccentric distance sum, Discrete Appl. Math. 160 (2012) 170–180. [6] A. Ilić, I. Gutman, Eccentric connectivity index of chemical trees, MATCH Commun. Math. Comput. Chem. 65 (2011) 731–744. [7] A. Ilić, G.H. Yu, L.H. Feng, On the eccentric distance sum of graphs, J. Math. Anal. Appl. 381 (2011) 590–600. [8] S. Li, Sharp bounds on the eccentric distance sum of graphs, in: I. Gutman, B. Furtula, K.C. Das, E. Milovanović, I. Milovanović (Eds.), Bounds in Chemical Graph Theory—Mainstreams, Univ. Kragujevac, Kragujevac, 2017, pp. 207–237. [9] 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. [10] L.Y. Miao, Q.Q. Cao, N. C, S.Y. Pang, On the extremal values of the eccentric distance sum of trees, Discrete Appl. Math. 186 (2015) 199–206. [11] M.J. Morgan, S. Mukwembi, H.C. Swart, A lower bound on the eccentric connectivity index of a graph, Discrete Appl. Math. 160 (2012) 248–258. [12] O. Ore, Theory of graphs, Amer. Math. Soc. Colloq. Publ. 38 (1962). [13] 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. [14] K. Xu, Ch. Das, A.D. Maden, On a novel eccentricity–based invariant of a graph, Acta Math. Sin. (Engl. Ser.) 32 (2016) 1477–1493. [15] K. Xu, K.C. Das, H. Liu, Some extremal results on the connective eccentricity index of graphs, J. Math. Anal. Appl. 433 (2016) 803–817. [16] K.X. Xu, L.H. Feng, Extremal energies of trees with a given domination number, Linear Algebra Appl. 435 (2011) 2382–2393. [17] G.H. Yu, L.H. Feng, On the connective eccentricity index of graphs, MATCH Commun. Math. Comput. Chem. 69 (2013) 611–628. [18] 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. [19] G. Yu, H. Qu, L. Tang, L. Feng, On the connective eccentricity index of trees and unicyclic graphs with given diameter, J. Math. Anal. Appl. 420 (2014) 1776–1786. [20] J. Zhang, B. Zhou, Z. Liu, On the minimal eccentric connectivity indices of graphs, Discrete Math. 312 (2012) 819–829.
Please cite this article in press as: L. Miao, et al., On the extremal values of the eccentric distance sum of trees with a given domination number, Discrete Applied Mathematics (2017), http://dx.doi.org/10.1016/j.dam.2017.04.032.