Bounds for the augmented Zagreb and the atom-bond connectivity indices

Applied Mathematics and Computation 307 (2017) 141–145

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Bounds for the augmented Zagreb and the atom-bond connectivity indices José Luis Palacios Electrical and Computer Engineering Department The University of New Mexico, Albuquerque, NM 87131, USA

a r t i c l e

i n f o

a b s t r a c t

Keywords: Atom-bond connectivity index Augmented Zagreb index Majorization Foster’s theorem

Using electrical networks and majorization we obtain a lower bound for the augmented Zagreb index in terms of the number of vertices and edges, and the maximum vertex degree. An analogous upper bound for the atom-bond connectivity index is also obtained. These bounds, which are attained by the complete graph Kn , are not comparable to the ones found in the literature. © 2017 Elsevier Inc. All rights reserved.

1. Introduction Let G = (V, E ) be a finite simple connected graph with vertex set V = {1, 2, . . . , n}, edge set E and degrees  = d1 ≥ d2 ≥ · · · ≥ dn = δ (For all graph theoretical terms the reader is referred to reference [24]). The ABC index, proposed by Estrada et al. in [8], and reintroduced in [9] was defined as





ABC (G ) =

(i, j )∈E

di + d j − 2 . di d j

(1)

The index ABC(G) has been studied in a large number of references of which we mention [6,11,13,14,16] and [18] for their own interest and for many other related references found in them. The Augmented Zagreb Index of G, defined by

AZI (G ) =

 (i, j )∈E



di d j di + d j − 2

3

,

(2)

was introduced by Furtula et al. in [12] as an alternative to the ABC index, with a better predictive power for heat of formation in several compounds. In [12] and [17] a number of properties of this index were proven, among them several lower and upper bounds further improved in [23]. In this note we use a couple of properties of electrical networks and the majorization technique in order to prove the inequalities in the abstract, which are not comparable to other similar bounds found in the literature. Here is a brief summary of majorization: given two n-tuples x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) with x1 ≥ x2 ≥ · · · ≥ xn and y1 ≥ y2 ≥ . . . ≥ yn , we say that x majorizes y and write xy in case k  i=1

xi ≥

k 

yi ,

i=1

E-mail address: [email protected] http://dx.doi.org/10.1016/j.amc.2017.03.009 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.

(3)

142

J.L. Palacios / Applied Mathematics and Computation 307 (2017) 141–145

for 1 ≤ k ≤ n − 1 and n 

n 

xi =

i=1

yi .

(4)

i=1

A Schur-convex function  : R → R keeps the majorization inequality, that is, if  is Schur-convex then xy implies (x) ≥ (y). Likewise, a Schur-concave function reverses the inequality: for this type of function xy implies (x) ≤ (y). A simple way to construct a Schur-convex (resp. Schur-concave) function is to consider

(x ) =

n 

f ( xi ),

i=1

where f : R → R is a convex (resp. concave) one-dimensional real function. For more details on majorization the reader is referred to [1–3,5] and [19]. Let Rij be the effective resistance between the vertices i and j found using Ohm’s law (that is, let Rij be the voltage at i when a battery is set between i and j so that the voltage at j is 0 and the current entering at i is 1, or alternatively, let Rij be the inverse of the current flowing between i and j when a unit volt battery is placed between i and j so that the voltage at j is 0). The following are well known facts that we will use below: (i) Foster’s theorem (see [10])



Ri j = n − 1.

(i, j )∈E

(ii) For (i, j) ∈ E we have (see [20])

1 ≥ Ri j ≥

di + d j − 2 2 ≥ . di d j − 1 n

For more details on the connection between electrical networks and chemical descriptors the reader may check references [21] and [22]. 2. The bounds The main ideas of what follows can be traced back to those used in [4] and [25]. We present the following results, found in section 2.3 of [3] (corollary 2.3.2 and theorem 2.3.2) as a lemma which will be used below  Lemma 1. Let  a be the set of real n-tuples x = (x1 , x2 , . . . , xn ) such that x1 ≥ x2 ≥ · · · ≥ xn and ni=1 xi = a. Let Sa be the set of n-tuples belonging to  a which additionally satisfy M ≥ xi ≥ m. Then





(i) The minimal element of  a is na , na , . . . na ; (ii) If the minimal element in (i) belongs to Sa , then it is also the minimal element of Sa ; −nm (iii) the maximal element of Sa is (M, M, . . . , M, θ , m, m, . . . , m ), where M appears k times, m appears n − k − 1 times, k =  aM−m  and θ = a − Mk − m(n − k − 1 ). We treat first the AZI index. Proposition 1. For any n-vertex G we have



AZI (G ) ≥

2 2 − 1

3

|E |4

( n − 1 )3

.

(5)

This bound is attained by the complete graph Kn . Proof. On account of fact (ii) in the introduction we have

AZI (G ) =

 (i, j )∈E

 ≥



di d j di + d j − 2

2 2 − 1

3

 (i, j )∈E

3

=

 (i, j )∈E



di d j di d j − 1

3 

di d j − 1 di + d j − 2

3

1 . R3i j

Now we will bound the last summation using majorization. Let us consider the subsets of R|E | defined as

n−1 = {x ∈ R|E | : x1 ≥ x2 ≥ · · · ≥ x|E | ;

|E |  j=1

x j = n − 1},

J.L. Palacios / Applied Mathematics and Computation 307 (2017) 141–145

143

and Sn−1 the subset of n−1 such that 1 ≥ xi ≥ 2n for 1 ≤ i ≤ |E|. By lemma 1 we can find explicitly the minimal element of Sn−1 , that is, a |E|-tuple x∗ such that xx∗ for x ∈ Sn−1 , indeed

x∗ =

n − 1 n − 1

n−1 , ,..., . |E | |E | |E |

Also, since f (x ) =

1 x3

(x ) ≥ (x∗ ) =

is convex, then (x ) =

|E |4

( n − 1 )3

|E |

1 i=1 x3 i

is Schur-convex, and

,

and since the |E|-tuple of effective resistances over the edges of the graph, properly arranged in decreasing order, belongs to the set Sn−1 on account of facts (i) and (ii), we are done proving (5). In the case of the complete graph Kn ,  = n − 1, |E | = n(n2−1 ) and a bit of algebra shows that both the bound and the n (n−1 )7 16(n−1 )3

actual value AZI(Kn ) equal



Remarks. In case the graph is a tree T, then |E | = n − 1 and (5) becomes



AZI (T ) ≥

2 2 − 1

3

(n − 1 ) ≥

( n − 1 )7 . n3 ( n − 2 )3

This bound, though giving the right order of magnitude, is weaker than those found in [12,17] and [23] for general and chemical trees. On the other hand, if we consider the n-vertex graph G1 to be a complete Kn−1 to which we add a new vertex v and two new edges vw and vz, where w and z are two different vertices of the Kn−1 , then our bound becomes

  4 (n − 1 )3 n−1 +2 2 AZI (G1 ) ≥ n3 ( n − 2 )3

which is roughly of order n5 , whereas the lower bound given in theorem 2.8 of [23] becomes for this graph



δ2 AZI (G1 ) ≥ |E | 2δ − 2

3





n−1 =2 2

+ 4.

Also, the lower bound given in theorem 2.3 of [23] is not attained by Kn , as ours does. Therefore, these remarks show that our bound is not comparable to those in the cited references. Now we deal with the ABC index. Proposition 2. For any n-vertex G we have



ABC (G ) ≤ where

 k=

  2 − 1 2 k + θ + ( |E | − k − 1 ) , n 2

n2 − n − 2|E | n−2

 and

(6)

2 n

θ = n − 1 − k − ( |E | − k − 1 ).

This bound is attained by the complete graph Kn , and the parenthesis in (6) is to be taken as n − 1 when the graph is a tree. Proof. On account of fact (ii) in the introduction we have

ABC (G ) =

 (i, j )∈E

 ≤



 di + d j − 2 = di d j

(i, j )∈E

2 − 1  Ri j . 2 (i, j )∈E



di d j − 1 di d j



di + d j − 2 di d j − 1

 In case the graph is a tree, all effective resistances are Ri j = 1 when (i, j) ∈ E, and so (i, j )∈E Ri j = n − 1. For all other cases, we will bound this summation using majorization. Let us consider again the set Sn−1 used in Proposition 1. By Lemma 1 we can identify explicitly the maximal element of Sn−1 , that is, the |E|-tuple x∗ such that x∗ x for all x ∈ Sn−1 , indeed



x∗ = 1, 1, . . . , 1, θ ,



2 2 2 , ,..., , n n n

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J.L. Palacios / Applied Mathematics and Computation 307 (2017) 141–145

where 1 appears k times,



k=

2 n



appears |E | − k − 1 times and

n2 − n − 2|E | , n−2

Since f (x ) =

2 n

θ = n − 1 − k − ( |E | − k − 1 ).

√  √ x is concave, then (x ) = m i=1 x is Schur-concave, and



(x ) ≤ (x ) = k + ∗



θ + ( |E | − k − 1 )

2 , n

and since the |E|-tuple of effective resistances over the edges of the graph, properly arranged in decreasing order, belongs to the set Sn−1 on account of facts (i) and (ii), we are done proving (5). In the case of the complete graph Kn we have  = n − 1, k = 0, θ = 2n and some algebra shows that both the bound √ n √n−2 2

and the actual value ABC(Kn ) equal



Remarks. The treatment of the ABC index given here differs from the one in [4], where ABC(G) was expressed in terms of the Randic´ index, and majorization was applied to this latter index, defined in terms of the degrees of G which add up to the constant 2|E|. The idea in the current article is to apply majorization to the effective resistances, which add up to the constant n − 1. The upper bound found in [4] is

ABC (G ) ≤ where R−1 (G ) =





(n − 1 )(|E | − R−1 (G )),

1 (i, j )∈E di d j

(7)

is the generalized Randic´ index with exponent −1. This bound is attained by both the complete

graph and the star graph Sn . In the case of a tree T, our new bound states that



ABC (T ) ≤

2 − 1 (n − 1 ) ≤ 2



( n − 1 )2 − 1 (n − 1 ) = n(n − 2 ), 2 (n − 1 )

so our new bound does not attain the value ABC (Sn ) =

(n − 1 )(n − 2 ).

On the other hand, for a -regular graph, the bound (7) becomes



 2 − 1  n (n − 1 ) , 2 2

ABC (G ) ≤

which is worse than our (6) in case



n (n − 1 )

 2

≥k+



 θ+

n −k−1 2



2 , n

(8)

(+1 ) ) where k =  n −nn−2  and θ = n − 1 − k − ( − 2 (k+1 n ). This is easy to achieve if, say,  = 3 and n is sufficiently large, because if that is the case, then the order of the left hand  2

side of (8) is roughly

3 2n

whereas the order of the right hand side is roughly n.

Thus, the comments above show that our new bound and (7) are not comparable. Moreover, the same examples show that our new bound is not comparable to another similar upper bound found in [15]: ABC (G ) ≤ |E |(n − 2R−1 (G )). Yet another upper bound in the literature, obtained in [7], states that



ABC (G ) ≤ p 1 −

1





+



[M1 (G ) − 2|E | − p(δ1 − 1 )] R−1 (G ) −

p





,

(9)

 where M1 (G ) = i∈V di2 is the first Zagreb index of G, p is the number of pendent vertices of and δ 1 is the minimal nonpendent degree. In case there are no pendent vertices, this upper bound becomes

ABC (G ) ≤



[M1 (G ) − 2|E |]R−1 (G ).

(10)

The bound (9) is attained by (1, )-semiregular graphs, specifically by Sn and, as we have mentioned above, our new bound does not attain ABC(Sn ). On the other hand, if we consider, for n large and a multiple of 3, the graph G∗ to be the symmetric barbell graph, composed of two copies of K n attached to each other through a linear graph of length n3 , then approximately, 3

|E | ∼

n2 9 ,

√

3

k ∼ 79 n, θ ∼ constant, and so asymptotically the bound (6) is equal to 92 n 2 . 2 3 1 Also, for this graph G∗ we have p = 0, M1 (G∗ ) ∼ 27 n and R−1 (G∗ ) ∼ 12 n, so that the bound (10) is asymptotically equal

 to

2 3 1 27 n 12 n

=

1 √ n2 . 9 2

This shows that our new bound and the one found in [7] are not comparable.

J.L. Palacios / Applied Mathematics and Computation 307 (2017) 141–145

145

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