Extremal graphs with respect to generalized ABC index

Discrete Applied Mathematics (

)

–

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Extremal graphs with respect to generalized ABC index Xiaodan Chen a , Guoliang Hao b, * a b

College of Mathematics and Information Science, Guangxi University, Nanning 530004, Guangxi, PR China College of Science, East China University of Technology, Nanchang, 330013, Jiangxi, PR China

article

a b s t r a c t

info

Article history: Received 21 August 2017 Received in revised form 30 November 2017 Accepted 20 January 2018 Available online xxxx

The generalized index of a graph G, denoted by ABCα (G), is defined as the sum of ( d +d −2 )ABC α over all edges vi vj of G, where α is an arbitrary non-zero real number, weights i d dj i j and di is the degree of vertex vi of G. In this paper, we first prove that the generalized ABC index of a connected graph will increase with addition of edge(s) if α < 0 or 0 < α ≤ 1/2, which provides a useful tool for the study of extremal properties of the generalized ABC index. By means of this result, we then characterize the graphs having the maximal ABCα value for α < 0 among all connected graphs with given order and vertex connectivity, edge connectivity, or matching number. Our work extends some previously known results. © 2018 Published by Elsevier B.V.

Keywords: Generalized ABC index Extremal graph Vertex connectivity Edge connectivity Matching number

1. Introduction Graph-based molecular structure descriptors, also known as topological indices, are numbers associated with chemical constitution for correlation of chemical structure with various physical properties, chemical reactivity or biological activity, which play an important role in Mathematical Chemistry, especially in the quantitative structure–property relationship (QSPR) and quantitative structure–activity relationship (QSAR) studies. Let G be a simple graph of order n with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G). The atom-bond connectivity index (or ABC index for short) of G, introduced by Estrada et al. [10] as a topological index, is defined by the sum of weights √ (di + dj − 2)/(di dj ) over all edges vi vj of G, namely,

√ ABC = ABC (G) =

∑

di + dj − 2

vi vj ∈E(G)

di dj

,

where di is the degree of vertex vi of G. The ABC index turned out to be closely correlated with the heat of formation of alkanes [10], and a quantum-chemical explanation for its descriptive ability was provided in [8]. Gutman et al. [22] later confirmed that the ABC index could reproduce the heat of formation with an accuracy comparable to that of high-level ab initio and DFT (MP2, B3LYP) quantum chemical calculations. Due to the above (chemical) applications, a large number of investigations on the mathematical properties, especially the extremal properties of the ABC index, have been triggered. Furtula et al. [12] proved that the star is the unique tree having the maximal ABC index among all trees with given order. The authors in [3,7] independently showed that the ABC index of a graph will increase with addition of edge(s). By means of this result, Chen et al. [4] characterized the graphs having the maximal ABC index among all connected graphs with given order and vertex connectivity. They also characterized the graph

*

Corresponding author. E-mail addresses: [email protected] (X. Chen), [email protected] (G. Hao).

https://doi.org/10.1016/j.dam.2018.01.013 0166-218X/© 2018 Published by Elsevier B.V.

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having maximal ABC index among all connected graphs with given order and matching number. However, unfortunately, their proofs are incorrect. We will point out the error in the end of Section 3. Recently, in [28] Zhang et al. studied the maximal ABC index among all connected graphs with given order and other parameters such as independence number, number of pendant vertices, edge connectivity, or chromatic index. In comparison with the studies of maximal properties, characterizing the trees having the minimal ABC index seems to be more difficult [20], and has been coined as the ‘‘ABC index conundrum’’ [19], see [14,18,24] for more information. For other researches on the ABC index one may also refer to [6,11,15,16,26,27,29]. On the other hand, in order to explore the better correlation abilities of the ABC index for the heat of formation of alkanes, Furtula et al. [13] made a generalization of this index by replacing 1/2 with an arbitrary non-zero real number α , i.e.,

∑ ( di + dj − 2 )α

ABCα = ABCα (G) =

di dj

vi vj ∈E(G)

,

and showed that ABC−3 has a better prediction power than the ABC index when studying the heat of formation of octanes and heptanes, which was named the augmented Zagreb index (abbreviated as AZI). They also proved that the star is the unique tree having the minimal AZI index among all trees with given order (recall, at the same time, that the star is the unique tree with the maximal ABC index). It seems very likely that the AZI and ABC indices have the opposite extremal properties, since their exponentials have the opposite signs. However, somewhat surprisingly, Huang et al. [23] proved that the AZI index of a connected graph would also increase with addition of edge(s), by which Ali et al. [1] further characterized the graphs having the maximal AZI index among all connected graphs with given order and vertex connectivity, or matching number. Note that in these cases the AZI and ABC indices have the same extremal properties. For more information on the significance of studying the AZI and ABC indices, one may refer to [17,21]. Recently, Estrada [9] referred to the above generalization ABCα as the generalized ABC index, and provided a probabilistic interpretation that fits very well with the chemical intuition for understanding the capacity of ABC -like indices to describe the energetics of alkanes. This work would also allow further investigations of more general scenarios outside molecular sciences, such as the study of random walks on graphs. In this paper, we are concerned with the extremal properties of the generalized ABC index of connected graphs. We first show that the generalized ABC index of a connected graph will increase with addition of edge(s) if α < 0 or 0 < α ≤ 1/2, which is an extension of the results in [3,7] for the ABC index and in [23] for the AZI index. This turns out to be a useful tool for the study of extremal properties of the generalized ABC index. By means of this result, we then characterize the graphs having the maximal ABCα value for α < 0 among all connected graphs with given order and vertex connectivity, edge connectivity, or matching number, which are extensions of the results in [1] for the AZI index. 2. Preliminaries We start this section by introducing some notation and terminology. Let Kn and K1,n−1 , as usual, denote the complete graph and the star with n vertices, respectively. Let Ka,b (a + b = n) be the complete bipartite graph with two partite sets having a and b vertices, respectively. Denote by G ∪ H the vertex-disjoint union of two graphs G and H. In particular, kG stands for the vertex-disjoint union of k copies of G. Let G ∨ H be the graph obtained from G ∪ H by adding all possible edges joining the vertices in G with those in H. For a vertex vi ∈ V (G), let NG (vi ) denote the set of its neighbours in G. For a vertex subset X ⊆ V (G), denote by G[X ] the subgraph of G induced by X and by G − X the subgraph of G induced by V (G) \ X , i.e., G − X = G[V (G) \ X ]. The vertex connectivity of a graph G, denoted by κ (G), is the minimum number of vertices whose removal from G yields a disconnected graph or a trivial graph, while the edge connectivity of G, denoted by κ ′ (G), is the minimum number of edges whose removal from G yields a disconnected graph. A matching in a graph G is a set of its disjoint edges, and the matching number of G, denoted by β (G), is the maximum cardinality of a matching over all its possible matchings. A connected component of a graph is said to be odd (resp., even) if it has an odd (resp., even) number of vertices. Let o(G) denote the number of odd components in the graph G. By the celebrated Tutte–Berge formula (see, for example, [25]), we have n − 2β (G) = max o(G − S) − |S | : S ⊆ V (G) .

{

}

(2.1)

We also need the following lemmas that will be used in the next section. Lemma 2.1. For any real number λ > 0, the function

[( f (x) = (x − 1)

x+1 x

)λ

( −

x

)λ ]

x−1

is strictly increasing with respect to x when x ≥ 2. Please cite this article in press as: X. Chen, G. Hao, Extremal graphs with respect to generalized ABC index, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.013.

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Proof. By a routine calculation, we have

( x )λ − x [ x−1 ] ( x + 1 )λ ( x )λ [( x + 1 )λ ( x )λ ] λ + + −x − x(x + 1) x x−1 x x−1 ( x+1 )λ [ ] 2 [ ( ) ] [ ( x x2 )λ ] λ x λ 1+ 2 + x(x + 1 − λ) 1 − 2 = x(x + 1) x −1 x −1 ( x+1 )λ [ ] 2 [( x )λ ] x > 2λ − x(x + 1 − λ) −1 . x(x + 1) x2 − 1

f ′ (x) =

( x + 1 )λ

(2.2)

Clearly, if x + 1 − λ ≤ 0, i.e., λ ≥ x + 1, then f ′ (x) > 0 and thus, the desired result follows. Therefore, in the following we may assume that 0 < λ < x + 1. To obtain the desired result, in view of (2.2), we just need to show that

[

2λ − x(x + 1 − λ)

] ( x2 ) λ − 1 ≥ 0, x2 − 1

that is,

)λ

x2

(

x(x + 1 − λ) + 2λ

≤

x2 − 1

x(x + 1 − λ)

,

which is equivalent to ln x(x + 1 − λ) + 2λ − ln x(x + 1 − λ)

(

λ≤

)

(

ln(x2 ) − ln(x2 − 1)

) .

(2.3)

Note( that, by the ) Lagrange Mean Value Theorem, there exist two real numbers ξ ∈ x(x + 1 − λ), x(x + 1 − λ) + 2λ and η ∈ x2 − 1, x2 such that

(

ln x(x + 1 − λ) + 2λ − ln x(x + 1 − λ)

(

)

ln(x2 )

−

(

ln(x2

)

− 1)

= 2λ

)

η 2λ(x2 − 1) ≥ . ξ x(x + 1 − λ) + 2λ

Furthermore, it is easy to check that for x ≥ 2 2λ(x2 − 1) x(x + 1 − λ) + 2λ

≥ λ,

implying that (2.3) is always obeyed when x ≥ 2 and hence, the desired result holds as well. The proof is completed. □ Lemma 2.2. For any real number λ > 0, 2 ( 23 )λ − 2λ + ( 94 )λ > 0.

]

[

Proof. Clearly, it suffices to prove that 1

>

2

( )λ 8

( )λ 2

−

9

3

( 9 )λ 4

[ ( )λ ] > 2 2λ − 23 , that is,

.

(2.4)

Note first that, if λ ≥ 6, a direct calculation shows that

( )λ 8

9

( )λ −

2

<

3

( )λ 8

9

( )6 8

≤

9

≈ 0.4933 <

1 2

.

Thus, in the following we just need to prove that (2.4) holds for ( )0x < (λ < )x 6 For 0 ≤ x ≤ 6, consider the (continuous) function h(x) = 98 − 23 and its first and second derivatives: ′

h (x) = h′′ (x) =

( )x [ 2

3

ln(8/9)

( )x 4

]

− ln(2/3) , ( )x ] )2 4 ( )2 ln(8/9) − ln(2/3) . 3

( )x [ ( 2 3

3

Please cite this article in press as: X. Chen, G. Hao, Extremal graphs with respect to generalized ABC index, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.013.

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X. Chen, G. Hao / Discrete Applied Mathematics ( ln(2/3) ln ln(8/9) ln(4/3)

(

Note that the equation h′ (x) = 0 has a unique root θ = 5.0’, it is not difficult to check that for 0 ≤ x ≤ 6, ln(8/9)

(

( ) )2 4 x 3

)

–

)

≈ 4.2971. Moreover, in virtue of software ‘Mathematica

( ) ( )2 ( )2 4 6 ( )2 − ln(2/3) ≤ ln(8/9) − ln(2/3) ≈ −0.0865, 3

which implies that h′′ (x) < 0. Consequently, when 0 ≤ x ≤ 6, the function h(x) attains its maximum value at x = θ and thus, h(x) ≤ h(θ ) ≈ 0.4277 <

1

,

2 as desired, completing the proof. □ Lemma 2.3. Let Φ1 (x) = 3x4 + (10k − 19)x3 + (11k2 − 46k + 43)x2 + (4k3 − 29k2 + 62k − 39)x − 2(k − 1)(k − 2)(k − 3). If k ≥ 2 and x ≥ 1, then Φ1 (x) ≥ 0. Proof. We shall prove that Φ1 (x) is a monotone increasing function with respective to x when k ≥ 2 and x ≥ 1, from which one may conclude that

Φ1 (x) ≥ Φ1 (1) = 2k(k − 1)(k − 2) ≥ 0. To this end, we consider the first three derivatives of Φ1 (x) with respective to x. For k ≥ 2 and x ≥ 1, we have

Φ1′′′ (x) = 72x + 60k − 114 > 0, implying that Φ1′′ (x) is monotonically increasing and thus,

Φ1′′ (x) ≥ Φ1′′ (1) = 22k2 − 32k + 8 > 0, again implying that Φ1′ (x) is monotonically increasing and consequently,

Φ1′ (x) ≥ Φ1′ (1) = 4k3 − 7k + 2 > 0, as desired, completing the proof. □ Lemma 2.4. For any real number α < 0, let Ψ1 (x) = (i) If k ≥ 2 and 1 ≤ x ≤

n−k , 2

x(x−1) 2

[ 2(x+k−2) ]α (x+k−1)2

, Ψ2 (x) = kx

[

α x+k+n−4 , (x+k−1)(n−1)

]

and Ψ (x) = Ψ1 (x) + Ψ2 (x).

then

Ψ (x) + Ψ (n − k − x) ≤ Ψ (1) + Ψ (n − k − 1). with equality if and only if x = 1. k (ii) If k = 1 and 2 ≤ x ≤ n− , then 2

Ψ (x) + Ψ (n − 1 − x) ≤ Ψ (2) + Ψ (n − 3) < Ψ (1) + Ψ (n − 2). k Proof. We first prove Part (i). For α < 0, we shall prove that Ψ ′′ (x) > 0 if k ≥ 2 and 1 ≤ x ≤ n− , from which it follows that 2 ′ Ψ (x) is strictly increasing with respective to x. Furthermore, noting that x ≤ n − k − x, we have

( )′ Ψ (x) + Ψ (n − k − x) = Ψ ′ (x) − Ψ ′ (n − k − x) ≤ 0, which implies that Ψ (x) + Ψ (n − k − x) reaches its maximum value at x = 1 and consequently, the desired result follows. Indeed, by a routine calculation, we have

]α [

] α 2 x(x − 1)(x + k − 3)2 − α Φ1 (x) Ψ1 (x) = +1 , (x + k − 1)2 2(x + k − 1)2 (x + k − 2)2 [ ]α x+k+n−4 α k(n − 3)Φ2 (x) Ψ2′′ (x) = , (x + k − 1)(n − 1) (x + k − 1)2 (x + k + n − 4)2 ′′

[

2(x + k − 2)

(2.5) (2.6)

where Φ1 (x) is defined as in Lemma 2.3 and

[ ] Φ2 (x) = (n − 3)α − n − 2k + 5 x − 2(n + k − 4)(k − 1). Bearing in mind that α < 0, k ≥ 2, x ≥ 1, and n ≥ 2x + k ≥ 4, we can easily check that Ψ1′′ (x) > 0 and Ψ2′′ (x) > 0 and hence, Ψ ′′ (x) > 0, as desired. k We next prove Part (ii). If k = 1 and 2 ≤ x ≤ n− , then from (2.5) and (2.6) it follows that 2

]α [

] α 2 (x − 2)2 − α (3x2 − 6x + 2) Ψ1 (x) = +1 , x2 2x(x − 1) [ ]α x + n − 3 α (α − 1)(n − 3)2 Ψ2′′ (x) = . (n − 1)x x(x + n − 3)2 ′′

[

2(x − 1)

Please cite this article in press as: X. Chen, G. Hao, Extremal graphs with respect to generalized ABC index, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.013.

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Noting that α < 0, x ≥ 2, and n ≥ 2x + k ≥ 5, we can also easily verify that Ψ1′′ (x) > 0 and Ψ2′′ (x) > 0 and hence, Ψ ′′ (x) > 0. Consequently, by the same arguments as in Part (i), we have Ψ (x) + Ψ (n − 1 − x) ≤ Ψ (2) + Ψ (n − 3). Now, it remains to be shown that Ψ (2) + Ψ (n − 3) < Ψ (1) + Ψ (n − 2). Indeed, setting α = −λ and by a direct calculation, we obtain

Ψ (1) + Ψ (n − 2) − Ψ (2) − Ψ (n − 3) [ ]λ [ ]λ (n − 2)(n − 3) (n − 2)2 (n − 3)(n − 4) (n − 3)2 = − 2 2(n − 3) 2 2(n − 4) [ ]λ ( )λ ( )λ (n − 1)(n − 2) 2 n−1 + (n − 2) − (n − 3) + − 3 × 2λ 2n − 5 n−1 n−2 ( )λ [ (n − 3)(n − 4) (n − 2)(n − 3) (n − 4)(n − 2)2 − = 2λ 2 (n − 3)3 2 ( )λ ( )λ ] 2n − 4 n−1 + (n − 2) − (n − 3) + −3 2n − 5 2(n − 2) [ ( )λ ] n − 1 > 2λ n − 5 + >0 2(n − 2) for λ > 0 and n ≥ 5 (since (n − 4)(n − 2)2 > (n − 3)3 and 2n − 4 > 2n − 5), as desired. This completes the proof. □

3. Main results Theorem 3.1. Let G be a connected graph with nonadjacent vertices vi and vj . If α < 0 or 0 < α ≤ 1/2, then ABCα (G + vi vj ) > ABCα (G). Proof. By the definition of the generalized ABC index, we have ABCα (G + vi vj ) − ABCα (G)

=

∑ [( di + dk − 1 )α ( di + dk − 2 )α ] − (di + 1)dk di dk vk ∈NG (vi ) ∑ [( dj + dk − 1 )α ( di + dk − 2 )α ] − + (dj + 1)dk di dk vk ∈NG (vj ) [ ]α di + dj + . (di + 1)(dj + 1)

(3.1)

Note that di + dk − 1 (di + 1)dk

−

di + dk − 2 di dk

{ > 0, = 0, =− (di + 1)di dk < 0,

if dk = 1 if dk = 2 if dk ≥ 3.

dk − 2

Therefore, if dk = 2, then

(

di + dk − 1

)α

( −

(di + 1)dk

di + dk − 2

)α = 0;

di dk

(3.2)

otherwise, from the Lagrange Mean Value Theorem it follows that

(

di + dk − 1

)α −

(di + 1)dk where ξ ∈

( di +dk −2 di dk

(

di + dk − 2

)α

di dk

= −αξ α−1

dk − 2 (di + 1)di dk

,

(3.3)

) ( d +dk −1 di +dk −2 ) k −1 , d(di ++d1)d if dk = 1, or ξ ∈ (di +1)d , dd if dk ≥ 3. i

k

i

k

i k

Suppose first that 0 < α ≤ 1/2. By (3.3), if dk = 1, then

(

di + dk − 1 (di + 1)dk

)α

( −

di + dk − 2 di dk

)α

> 0;

(3.4)

Please cite this article in press as: X. Chen, G. Hao, Extremal graphs with respect to generalized ABC index, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.013.

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if dk ≥ 3, then

(

di + dk − 1

)α

( −

(di + 1)dk

> −α

(

di + dk − 1

di + dk − 2

)α

di dk

)α−1

(di + 1)dk

dk − 2 (di + 1)di dk

)α di + dk − 1 dk − 2 −α di (di + 1)α dk di + dk − 1 ( ) −α dk − 2 −α ≥ > . α di (di + 1) dk di (di + 1)α (

=

(3.5)

Without loss of generality, assume that 1 ≤ di ≤ dj . Observing that ⏐{vk : vk ∈ NG (vi ), dk ≥ 3}⏐ ≤ di , and combining (3.1) and (3.2), (3.4) and (3.5), we obtain

⏐

ABCα (G + vi vj ) − ABCα (G)

[(

∑

≥

di + dk − 1

[(

∑

dj + dk − 1

+

di + dk − 2

)α ]

di dk

)α

( −

(dj + 1)dk

vk ∈NG (vj ), dk ≥3

[

( −

(di + 1)dk

vk ∈NG (vi ), dk ≥3

+

)α

⏐

di + dk − 2

)α ]

di dk

]α

di + dj (di + 1)(dj + 1)

[ ]α −α di + dj −α + + (di + 1)α (dj + 1)α (di + 1)(dj + 1) ( )α −2α 1 di + dj ≥ + ≥ 0, (di + 1)α (di + 1)α dj + 1 >

as desired. Suppose now that α < 0. If dk ≥ 3, then by (3.3), we have

(

di + dk − 1

)α

( −

(di + 1)dk

di + dk − 2

)α

di dk

> 0.

(3.6)

Since G is connected, if vk ∈ NG (vi ) and dk = 1, then di ≥ 2. Moreover, it is easy to see that ⏐{vk : vk ∈ NG (vi ), dk = 1}⏐ ≤ di −1. Now, combining (3.1) and (3.2) and (3.6), and setting α = −λ, we get

⏐

ABCα (G + vi vj ) − ABCα (G)

[(

∑

≥

di + dk − 1

[(

+

dj + dk − 1

+

( −

di + dk − 2

)α ]

di dk

]α

di + dj

≥ (di − 1)

di + 1

+ (dj − 1)

[( )λ 3

2

)λ

( −

di

[(

> 0,

)α ]

(di + 1)(dj + 1)

[(

≥2

di + dk − 2 di dk

)α

(dj + 1)dk

vk ∈NG (vj ), dk =1

[

( −

(di + 1)dk

vk ∈NG (vi ), dk =1

∑

)α

⏐

dj + 1

− 2λ +

di − 1

)λ

( −

dj

]

)λ ]

di dj

dj − 1

)λ ]

[ +

(di + 1)(dj + 1)

]λ

di + dj

( )λ 9

4 (by Lemma 2.2)

(by Lemma 2.1)

as desired, completing the proof. □ Remark 1. First, one can easily see that Theorem 3.1 extends some previously known results in [3,7] for the ABC index and in [23] for the AZI index. Please cite this article in press as: X. Chen, G. Hao, Extremal graphs with respect to generalized ABC index, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.013.

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Fig. 1. The graph K6 (2).

Second, from the above proof for the case of 0 < α ≤ 1/2, one might find that the upper bound 1/2 seems improvable by using some more subtle arguments. However, there is no much room to improve it. To see this, let us consider the star K1,n−1 with v1 , v2 being its two pendant vertices (i.e., vertices of degree one). By (3.1), we have ABCα (K1,n−1 + v1 v2 ) − ABCα (K1,n−1 )

( )α ( )α ( )α [ ( )α ] 1 n−2 1 2(n − 2) =3 −2 = 3−2 . 2 n−1 2 n−1

If α >

ln 3 ln 2

− 1 ≈ 0.585, then 2 −

( 3 ) α1 2

( )1

> 0 and thus, for n >

4− 32

α

( )1 α 2− 32

, we have 3 − 2

( 2(n−2) )α n−1

< 0, i.e., ABCα (K1,n−1 + v1 v2 ) <

ABCα (K1,n−1 ). Finally, a question one may ask naturally is whether or not ABCα (G + vi vj ) < ABCα (G) holds for all connected graphs G 3 − 1? The following result, which can be deduced from (3.1) directly, gives a with sufficiently many vertices when α > ln ln 2 negative answer to this question. Proposition 3.2. Let G be a connected graph with two pendant vertices vi and vj , and vs and vt be the (unique) adjacent vertices of vi and vj , respectively. If dG (vs ) = dG (vt ) = 2, then for any non-zero real number α , ABCα (G + vi vj ) > ABCα (G). As direct consequences of Theorem 3.1, we have the following results. Corollary 3.3. Among all connected graphs with n ≥ 3 vertices, the complete ( ) graph Kn is the unique graph that has the maximal n(n−1) 2n−4 α . ABCα value for α < 0 or 1 < α ≤ 1/2. Moreover, ABCα (Kn ) = 2 2 (n−1) Corollary 3.4. Among all connected bipartite graphs with n ≥ 3 vertices, the complete bipartite graph K⌊n/2⌋,⌈n/2⌉ is the unique

)1−α

graph that has the maximal ABCα value for α < 0 or 1 < α ≤ 1/2. Moreover, ABCα K⌊n/2⌋,⌈n/2⌉ = ⌊n/2⌋⌈n/2⌉

(

)

(

(n − 2)α .

Corollary 3.5. Among all connected graphs with n ≥ 3 vertices, the graph that has the minimal ABCα value for α < 0 or 1 < α ≤ 1/2 must be a tree. Furthermore, by means of Theorem 3.1, we can characterize the graphs having the maximal generalized ABC index among all connected graphs with given order and vertex connectivity, edge connectivity, ) or matching number. ( n(n−1) 2n−4 α Recall that if κ (G) = n − 1, then G = Kn , and ABCα (Kn ) = . So, in the following we may assume that 2 (n−1)2 1 ≤ κ (G) ≤ n − 2. For 1 ≤ k ≤ n − 2, let Kn (k) = Kk ∨ (K1 ∪ Kn−k−1 ), which is the graph obtained by joining one vertex to k vertices in Kn−1 . An illustration of K6 (2) is shown in Fig. 1. Theorem 3.6. Among all connected graphs with n ≥ 5 vertices and vertex connectivity not more than k (1 ≤ k ≤ n − 2), the graph Kn (k) is the unique graph that has the maximal ABCα value for α < 0. Moreover, ABCα Kn (k) =

(

[ n+k−3 ]α

k

(n−1)k

+

[ (n−k−1)(n−k−2) 2

2n−6 (n−2)2

]α

+ k(n − k − 1)

[

2n−5 (n−1)(n−2)

]α

)

k(k−1) 2

[

2n−4 α (n−1)2

]

+

.

Proof. Note first that if)ℓ < k, then it) is easy to check that Kn (ℓ) is an edge-deleted subgraph of Kn (k) and hence, by ( ( Theorem 3.1, ABCα Kn (ℓ) < ABCα Kn (k) for α < 0. So, in the following we just need to consider those graphs with vertex connectivity equal to k, where 1 ≤ k ≤ n − 2. For α < 0, we suppose that G∗ is a graph having the maximal ABCα value among all connected graphs with n ≥ 5 vertices and κ (G∗ ) = k. By the definition of vertex connectivity, there exists a vertex subset X ⊂ V (G∗ ) with |X | = k such that G∗ − X is disconnected. Moreover, G∗ − X has exactly two connected components, say G∗1 and G∗2 . In fact, if G∗ − X has Please cite this article in press as: X. Chen, G. Hao, Extremal graphs with respect to generalized ABC index, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.013.

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at least three connected components, then inserting an edge between any two connected components would preserve the vertex connectivity of G∗ but increase the ABCα value (by Theorem 3.1), contradicting the maximality of G∗ . Also, by a similar reasoning, we may conclude that G∗1 , G∗2 , and G∗ [X ] are complete graphs, and each vertex of G∗1 and G∗2 is adjacent to each in X . Consequently, G∗ = Kk ∨ (Kn1 ∪ Kn2 ), where n1 = |V (G∗1 )| and n2 = |V (G∗2 )|. We may assume, without loss of generality, that k 1 ≤ n1 ≤ n2 . Clearly, n1 + n2 = n − k and n1 ≤ n− . It is also easy to see that di = n1 + k − 1 for vi ∈ V (G∗1 ), dj = n2 + k − 1 2 ∗ for vj ∈ V (G2 ), and dt = n − 1 for vt ∈ X . Now, by the definition of ABCα and noting that n2 = n − k − n1 , we have ∗

ABCα (G ) =

k(k − 1)

[

2n − 4

]α

(n − 1)2

2

[

+

n1 (n1 − 1) 2(n1 + k − 2) (n1 + k − 1)2

2

[

+ =

n2 (n2 − 1) 2(n2 + k − 2) (n2 + k − 1)2

2

k(k − 1)

[

2n − 4

]α

(n − 1)2

2

]α

[ + kn1

]α

(n1 + k − 1)(n − 1)

[ + kn2

n1 + k + n − 4 n2 + k + n − 4

]α ]α

(n2 + k − 1)(n − 1)

+ Ψ (n1 ) + Ψ (n − k − n1 ),

(3.7)

where Ψ (x) is defined as in Lemma 2.4. If k ≥ 2, then by (3.7) and Part (i) of Lemma 2.4, and bearing the maximality of G∗ in mind, we obtain ∗

ABCα (G ) =

=

k(k − 1)

[

2

]α

(n − 1)2

2 k(k − 1)

2n − 4

[

2n − 4 (n − 1)2

]α

+ Ψ (n1 ) + Ψ (n − k − n1 ) + Ψ (1) + Ψ (n − k − 1),

from which it follows that n1 = 1 and thus, G∗ = Kn (k). If k = 1, then by (3.7) and Part (ii) of Lemma 2.4, and noting that the maximality of G∗ , we have ABCα (G∗ ) = Ψ (n1 ) + Ψ (n − 1 − n1 ) = Ψ (1) + Ψ (n − 2), which implies that G∗ = Kn (1). Finally, the ABCα value of the graph Kn (k) follows directly from (3.7) by substituting n1 = 1. This completes the proof. □ It is well known that κ (G) ≤ κ ′ (G) for any connected graph G (see, for example, [2]), which implies ( ) that any connected graph G with κ ′ (G) ≤ k is also a connected graph with κ (G) ≤ k. Moreover, it is easy to see that κ ′ Kn (k) = k. Consequently, the next corollary follows immediately from Theorem 3.6. Corollary 3.7. Among all connected graphs with n ≥ 5 vertices and edge connectivity not more than k (1 ≤ k ≤ n − 2), the graph Kn (k) is the unique graph that has the maximal ABCα value for α < 0. Theorem 3.8. Among all connected graphs with n ≥ 5 vertices and matching number not more than β (1 ≤ β ≤ ⌊n/2⌋), (i) if β = ⌊n/2⌋, then the complete graph Kn is the unique graph that has the maximal ABCα value for α < 0. (ii) If 1 ≤ β (≤ ⌊n/2⌋ − 1, then graph unique ) the ( Kβ ∨) (n − β )K1 is (the ) graph that has the maximal ABCα value for α < 0. β (β−1) 2n−4 α n+β−3 α Moreover, ABCα Kβ ∨ (n − β )K1 = + β (n − β ) . 2 (n−1)β (n−1)2 Proof. The reviewer of this paper pointed out that this proof can be shortened by using Lemma 1.2 in [5] as its start point. But for the sake of completeness, we here would like to include the whole proof. As in the proof of Theorem 3.6, if ℓ < β , then ( it is easy to)verify that ( Kℓ ∨ (n − ℓ)K)1 is an edge-deleted subgraph of Kβ ∨ (n − β )K1 and thus, by Theorem 3.1, ABCα Kℓ ∨ (n − ℓ)K1 < ABCα Kβ ∨ (n − β )K1 for α < 0. So, in the following it suffices to consider those graphs with matching number equal to β , where 1 ≤ β ≤ ⌊n/2⌋. For α < 0, let G∗ be a graph having the maximal ABCα value among all connected graphs with n ≥ 5 vertices and β (G∗ ) = β . From (2.1) it would follow that there exists a vertex subset X ⊂ V (G∗ ) such that n − 2β = max{o(G∗ − S) − |S | : S ⊆ V (G∗ )} = o(G∗ − X ) − |X |. For convenience, let |X | = k and o(G∗ − S) = p. Clearly, n ≥ p + k. Moreover, since p − k = n − 2β , we have p ≥ k and 0 ≤ k ≤ β . If β = ⌊n/2⌋, then p − k ≤ 1. Observe that β (Kn ) = ⌊n/2⌋ and thus, by Corollary 3.3, we have G∗ = Kn , completing the proof of Part (i). If 1 ≤ β ≤ ⌊n/2⌋ − 1, then p − k ≥ 2. Moreover, we have k ≥ 1 (otherwise, G∗ − X = G∗ and o(G∗ − X ) = p ≤ 1, a contradiction), and hence p ≥ 3. Suppose that G∗1 , G∗2 , . . . , G∗p are all the odd components of G∗ − X . Note that G∗1 , G∗2 , . . . , G∗p are all the connected components of G∗ − X as well. Indeed, if G∗ − X has an even component, then by inserting an edge between the even component and any odd component we may obtain a new graph G∗∗ . Clearly, β (G∗∗ ) ≥ β (G∗ ) = β . On the other hand, by (2.1) and noting that o(G∗∗ − X ) = o(G∗ − X ), we have n − 2β (G∗∗ ) ≥ o(G∗∗ − X ) −|X | = o(G∗ − X ) −|X | = n − 2β , that is, β (G∗∗ ) ≤ β . Consequently, β (G∗∗ ) = β . But, from Theorem 3.1 it follows that ABCα (G∗∗ ) > ABCα (G∗ ), contradicting Please cite this article in press as: X. Chen, G. Hao, Extremal graphs with respect to generalized ABC index, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.013.

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9

the maximality of G∗ . Also, by a similar reasoning, we may conclude that G∗1 , G∗2 , . . . , G∗p and G∗ [X ] are complete graphs, and each vertex of G∗1 , G∗2 , . . . , G∗p is adjacent to each in X . Thus, G∗ = Kk ∨ (Kn1 ∪ Kn2 ∪ · · · ∪ Knp ), where nt = |V (G∗t )|, t = 1, 2, . . . , p. Clearly, n1 + n2 + · · · + np = n − k and n1 , n2 , . . . , np are odd numbers. We also assume, without loss of generality, that 1 ≤ n1 ≤ n2 ≤ · · · ≤ np . Now, by the definition of ABCα and observing that di = nt + k − 1 for vi ∈ V (G∗t ) and dj = n − 1 for vj ∈ X , we have ABCα (G∗ ) =

k(k − 1)

[

2n − 4

]α

(n − 1)2

2

[

+

n1 (n1 − 1) 2(n1 + k − 2) (n1 + k − 1)2

2

[

+

n2 (n2 − 1) 2(n2 + k − 2)

[

=

np (np − 1) 2(np + k − 2) (np + k − 1)2

2

k(k − 1)

[

2n − 4

]α

(n − 1)2

2

[ + kn1

]α

]α

[ + knp

]α

n1 + k + n − 4 (n1 + k − 1)(n − 1)

[ + kn2

(n2 + k − 1)2

2

+ ··· +

]α

n2 + k + n − 4

]α

(n2 + k − 1)(n − 1)

]α

np + k + n − 4 (np + k − 1)(n − 1)

+ Ψ (n1 ) + Ψ (n2 ) + · · · + Ψ (np ),

(3.8)

where Ψ (x) is defined as in Lemma 2.4. Note that if np−1 > 1, then np ≥ np−1 ≥ 3. Let np−1 + np + k = n′ . Then, np = n′ − k − np−1 and np−1 ≤ we have

n′ −k . By Lemma 2.4, 2

Ψ (np−1 ) + Ψ (np ) = Ψ (np−1 ) + Ψ (n′ − k − np−1 ) < Ψ (1) + Ψ (n′ − k − 1) = Ψ (1) + Ψ (np−1 + np − 1), which shows that when replacing np−1 and np by 1 and np−1 + np − 1 in (3.8) respectively (note that the resulting graph after this operation is Kk ∨ (Kn1 ∪ Kn2 ∪ · · · ∪ K1 ∪ Knp−1 +np −1 )), the ABCα value would increase strictly, contradicting the maximality of G∗ . Therefore, we have n1 = n2 = · · · = np−1 = 1, which implies that G∗ = Kk ∨ (p − 1)K1 ∪ Knp = Kk ∨ (n − 2β + k − 1)K1 ∪ K2β−2k+1 .

)

(

(

)

Furthermore, if k < β , then it is not difficult to see that the graph Kk ∨ (n − 2β + k − 1)K1 ∪ K2β−2k+1 is an edge-deleted subgraph of Kβ ∨ (n − β )K1 , and from Theorem 3.1 it follows that ABCα (G∗ ) < ABCα (Kβ ∨ (n − β )K1 ), again contradicting the maximality of G∗ . Consequently, we have k = β and G∗ = Kβ ∨ (n − β )K1 , as desired. In addition, the ABCα value of the graph Kβ ∨ (n − β )K1 follows directly from (3.8) by substituting k = β and n1 = n2 = · · · = np = 1. The proof is completed. □

)

(

Remark 2. Note that Theorems 3.6 and 3.8 extend the results in [1] for the AZI index. For 0 < α ≤ 1/2, we have not yet characterized the graphs having the maximal ABCα value among all connected graphs with given order and vertex connectivity, edge connectivity, or matching number. This seems to be complicated. In [4], the authors characterized those extremal graphs for the special case α = 1/2, but√ unfortunately, there√ was a mistake in the proof of their key lemma (i.e., Lemma 4). Specifically speaking, let f (x) =

x(x−1) 2

2k+2x−4 , (k+x−1)2

g(x) = kx

k+x+n−4 , (k+x−1)(n−1)

and

F (x) = f (x) + g(x), where k ≥ 1, x ≥ 1, and n − x − k ≥ 1. When k ≥ 2, in order to obtain the desired result, the authors in [4] needed to prove that F ′′ (x) > 0 and hence, they calculated f ′′ (x) and g ′′ (x). However, the expression of g ′′ (x) is wrong. In fact, by using software ‘Mathematica 5.0’, we have k(n − 3) g ′′ (x) = −

√

k+x+n−4 (k+x−1)(n−1)

[

4(k − 1)(n + k − 4) + (n + 4k − 7)x

4(k + x − 1)2 (k + x + n − 4)2

] < 0.

With this new expression, proving F ′′ (x) > 0 becomes a difficult problem. In our future work, we shall seek an effective method to solve this problem, and even to characterize those extremal graphs for the case 0 < α ≤ 1/2. Acknowledgements The authors would like to thank the anonymous referees for their helpful comments and suggestions towards improving the original version of this paper. This work was supported by National Natural Science Foundation of China (No. 11501133), Natural Science Foundation of Guangxi Province (Nos. 2016GXNSFAA380293, 2014GXNSFBA118008). Please cite this article in press as: X. Chen, G. Hao, Extremal graphs with respect to generalized ABC index, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.01.013.

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