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Correction to “Bounds for the augmented Zagreb and the atom-bond connectivity indices”
Applied Mathematics and Computation 362 (2019) 124559
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Correction to “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 b s t r a c t We provide upper bounds for the atom-bond connectivity index that correct some bounds given in [6], using the same majorization technique. © 2019 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 . A graph is d-regular if all its vertices have degree d; it is (d, 1)-semiregular if its vertices have either degree 1 or d. (For all graph theoretical terms the reader is referred to reference [7]). The ABC index, proposed by Estrada et al. in [4], and reintroduced in [5] was defined as
ABC (G ) =
(i, j )∈E
di + d j − 2 . di d j
In [6] we observed that
ABC (G ) =
(i, j )∈E
≤
di d j − 1 di d j
(1)
di + d j − 2 di d j − 1
2 − 1 Ri j , 2 (i, j )∈E
where Rij is the effective resistance between vertices i and j, computed with Ohm’s laws, when the graph is thought of as an electric network where the edges have unit resistance. In case the graph is a tree T, all effective resistances are Ri j = 1 when (i, j) ∈ E, and so we got the bound
ABC (T ) ≤
2 − 1 ( n − 1 ). 2
For all other cases, we tried to bound the summation of the roots of the effective resistances using majorization: since √ √ f (x ) = x is concave, then (x ) = m i=1 x is Schur-concave, and
(x ) ≤ (x∗ ), DOI of original article: 10.1016/j.amc.2017.03.009 E-mail address: [email protected] https://doi.org/10.1016/j.amc.2019.124559 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.
2
J.L. Palacios / Applied Mathematics and Computation 362 (2019) 124559
where x∗ is the minimal element of Sn−1 , the subset of
x ∈ R|E | : x1 ≥ x2 ≥ · · · ≥ x|E | ;
n−1 =
|E |
xj = n − 1 ,
j=1
consisting of those |E|-tuples such that 1 ≥ xi ≥ 2n for 1 ≤ i ≤ |E|. Unfortunately, we used the maximal x∗ instead of the minimal element, and thus all upper bounds for ABC(G) in [6], except those for trees, are not justified. If we consider the correct minimal element
x∗ =
n − 1 n − 1 n−1 , ,..., , |E | |E | |E |
we obtain that
ABC (G ) ≤
(2 − 1 )|E |(n − 1 ) . 2
From the fact that R−1 (G ) ≡
ABC (G ) ≤
|E | 1 (i, j )∈E di d j ≥ 2 , we can see that the bound so obtained is always worse than
(n − 1 )(|E | − R−1 (G )),
(2)
that was found in [1]. Thus, majorization on the effective resistances does not lead to better results in this context. 2. A different majorization Instead of looking at effective resistances, now we look at the quantities defined by
Ai j =
1 1 + , di dj
for all (i, j) ∈ E. It is clear that we can write
ABC (G ) ≤
2
Ai j −
2
(i, j )∈E
.
(3)
Now we can prove the following general upper bound: Proposition 1. For any n-vertex graph we have
ABC (G ) ≤ |E |
n
|E |
−
2
2
.
(4)
The equality is attained by any d-regular graph. Proof. The majorization of the summation in the right hand side of (3) is carried out by looking at the Schur-concave function
(x ) =
|E |
xi −
i=1
2
2
,
|E | on the set n of |E|-tuples x = (x1 , x2 , . . . , x|E | ) such that i=1 xi = n. Note that the Aij s belong to this set because of the general relation (i, j )∈E (diα + dαj ) = ni=1 diα +1 , which turns out the desired result when we take α = −1. Now we have that the minimal element of n is x∗ = ( |En| , |En| , . . . , |En| ), and thus
ABC (G ) ≤ ({Ai j } ) ≤ (x∗ ) = |E |
n
|E |
−
2
2
.
It is a simple calculation to verify that for any d-regular graph G the value of the bound and ABC(G) are both equal to n
d−1 2
Remarks 1. Bound (4) expressed only in terms of the number of vertices, the number of edges and the largest degree in the graph, seems to be new. It is clear that (4) and (2) are not comparable, by looking at those graphs where the equalities are attained. It should be remarked that (2) was derived without using majorization, although a similar majorization of the Aij s was used in [1] to tackle R−1 (G ), the Randic´ index, in bound (2). In recent literature some bounds similar to (4) can be found; specifically in [3] one finds that
+δ , δ
ABC (G ) < |E | √
(5)
J.L. Palacios / Applied Mathematics and Computation 362 (2019) 124559
and
ABC (G ) ≤ |E |
√ 2 − 2
δ
3
.
(6)
+δ ≥ 2, whereas It is clear that (4) is always better than (5) because by the arithmetic mean-geometric mean inequality √ δ
√ n 2 n n − < ≤ ≤ 2 , for n ≥ 2. n−1 |E | |E | 2
Also, if δ = , that is, if the graph is regular, (4) and (6) coincide, but for all other cases, (6) is weaker than (4). Indeed, for non-regular graphs we have
δ + 1 ≤ , implying
(δ + 1 )2 ≤ 3 < 3 + δ 2 , from which we get
δ 2 − δ 2 < 3 − 2 . Multiplying this inequality by δ 222 we obtain
2
δ
−
2
<
2
2 − 2
δ2
,
and taking square roots we get
2
δ
−
2
2
<
√ 2 − 2
δ
.
Finally, since 2|E| ≥ nδ , we have
2 n − ≤ | E | 2
2
δ
−
√
2
<
2
2 − 2
δ
.
We can also specify the number of pendant vertices, h, for finer results. Let us denote by E1 the set of pendant edges, that is, those edges one of whose end vertices has degree 1, and let us denote E2 = E − E1 . Also, if there are h pendant vertices, it is clear that dn−h is the smallest degree strictly larger than 1. With these notations we have the following: Proposition 2. For any n-vertex graph G with h pendant vertices we have
1
ABC (G ) ≤ h 1 +
dn−h
−
2
1
+ ( |E | − h )
n − h − h1 2 − 2, |E | − h 2
(7)
where i is the largest degree in Ei , i = 1, 2. The equality is attained by all d-regular graphs and all (d, 1)-semiregular graphs. Proof. It is clear that we can write
ABC (G ) =
(i, j )∈E
di + d j − 2 ≤ di d j
Ai j −
(i, j )∈E1
2
1
+
(i, j )∈E2
Ai j −
2
22
.
(8)
Now we will apply majorization separately on each summand. For the first summand, we have |E1 | = h, Ai j =
and dn−h ≤ d j ≤ 1 . We set (i, j )∈E1 Ai j = X, and consider the Schur-concave function 1 (x ) = |E1 | set of |E1 |-tuples such that i=1 xi = X. 1+
1 dj
The minimal element in this context is x∗ =
Ai j −
(i, j )∈E1
2
1
h 1+
1
1
2 X − . h 1
≤ (x∗ ) = h
Also, we can bound X as follows:
≤X ≤h 1+
1
X
h
, Xh , . . .
X h
|E1 | i=1
1 di
+
1 dj
=
xi − 2 on the 1
and we bound the first summand as follows:
(9)
,
(10)
2 1 − . dn−h 1
(11)
dn−h
and if we use (10) in (9), we get
(i, j )∈E1
Ai j −
2
1
≤h 1+
4
J.L. Palacios / Applied Mathematics and Computation 362 (2019) 124559
For the second summand in (8) we have |E2 | = |E | − h and Ai j = d1 + d1 with dn−h ≤ di , d j ≤ 2 . We set (i, j )∈E1 Ai j = n − X, i j
| E 2 | | E 2 | and consider the Schur-concave function 2 (x ) = i=1 xi − 22 on the set of |E2 |-tuples such that x = n − X. The i=1 i minimal element now is x∗ =
Ai j −
(i, j )∈E2
n−X n−X n−X |E |−h , |E |−h , . . . |E |−h
2
22
≤ (x∗ ) = (|E | − h )
2
and we bound the second summand as
n−X
|E | − h
−
2
22
.
(12)
Also, using (10), we can bound n − X as follows:
n−h−
h h ≤n−X ≤n−h− , dn−h 1
(13)
and if we insert (13) into (12), we get
Ai j −
(i, j )∈E2
2
22
≤ ( |E | − h )
n − h − h1 2 − 2. |E | − h 2
(14)
Inserting now (11) and (14) into (8) finishes the proof of (7). As for the equality in the case of (d, 1)-semiregular graphs, it is easy to see that in that case both (1) and (7) yield the value
1 h 1 − + ( |E | − h ) d
2 2 − 2, d d
once we notice that the sum of all degrees is 2|E | = (n − h )d + h
Remarks 2. Das found in [2] the following upper bound:
ABC (G ) ≤ h 1 −
1
+ ( |E | − h )
2 2 − 2 , dn−h dn−h
(15)
which resembles our bound (7) and also attains the equality for d-regular and (d, 1)-semiregular graphs. These bounds (15) and (7) are not comparable. Indeed, consider an n-cycle Cn to which we add n − 3 extra edges between a fixed vertex i in the cycle and all others to which it was not previously linked; now attach to every one of the vertices a copy of a 4-star graph through one of its leaves. The resulting graph Gn satisfies |V | = 4n, |E | = 5n − 3, h = 2n, d|V |−h = 1 = 3 and = 2 = di = n. Bound (15) produces
1 ABC (Gn ) ≤ 2n 1 − + (3n − 3 ) n whereas bound (7) gives
ABC (Gn ) ≤ 2n
2 + ( 3n − 3 ) 3
2 2 − ∼ 4n, 3 9
2n − 23n 2 − 2 ∼ 3.63n. 3n − 3 n
On the other hand, if we consider two n-cycles which are attached through a single vertex, this is a graph Hn satisfying
|V | = 2n − 1, |E | = 2n, h = 0, d|V |−h = 2, = 2 = 4. Bound (15) produces 2n ABC (Hn ) ≤ √ ∼ 1.41n, 2 whereas bound (7) turns out
ABC (Hn ) ≤ 2n
2 2n − 1 − ∼ 1.87n. 2n 16
References [1] [2] [3] [4]
M. Bianchi, A. Cornaro, J.L. Palacios, A. Torriero, New upper bounds for the ABC index, MATCH Commun. Math. Comput. Chem. 76 (2016) 117–130. K.C. Das, Atom-bond connectivity index of graphs, Discr. Appl. Math. 158 (2010) 1181–1188. K.C. Das, S. Elumalai, I. Gutman, On ABC index of graphs, MATCH Commun. Math. Comput. Chem. 78 (2017) 459–468. E. Estrada, L. Torres, L. Rodríguez, I. Gutman, An atom-bond connectivity index: modelling the enthalpy of formation of alkanes, Indian J. Chem. 37A (1998) 849–855. [5] E. Estrada, Atom-bond connectivity and the energetic of branched alkanes, Chem. Phys. Lett. 463 (2008) 422–425. [6] J.L. Palacios, Bounds for the augmented Zagreb and the atom-bond connectivity indices, Appl. Math. Comput. 307 (2017) 141–145. [7] R.J. Wilson, Introduction to Graph Theory, Oliver & Boyd, Edinburgh, 1972.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Correction to “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 b s t r a c t We provide upper bounds for the atom-bond connectivity index that correct some bounds given in [6], using the same majorization technique. © 2019 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 . A graph is d-regular if all its vertices have degree d; it is (d, 1)-semiregular if its vertices have either degree 1 or d. (For all graph theoretical terms the reader is referred to reference [7]). The ABC index, proposed by Estrada et al. in [4], and reintroduced in [5] was defined as
ABC (G ) =
(i, j )∈E
di + d j − 2 . di d j
In [6] we observed that
ABC (G ) =
(i, j )∈E
≤
di d j − 1 di d j
(1)
di + d j − 2 di d j − 1
2 − 1 Ri j , 2 (i, j )∈E
where Rij is the effective resistance between vertices i and j, computed with Ohm’s laws, when the graph is thought of as an electric network where the edges have unit resistance. In case the graph is a tree T, all effective resistances are Ri j = 1 when (i, j) ∈ E, and so we got the bound
ABC (T ) ≤
2 − 1 ( n − 1 ). 2
For all other cases, we tried to bound the summation of the roots of the effective resistances using majorization: since √ √ f (x ) = x is concave, then (x ) = m i=1 x is Schur-concave, and
(x ) ≤ (x∗ ), DOI of original article: 10.1016/j.amc.2017.03.009 E-mail address: [email protected] https://doi.org/10.1016/j.amc.2019.124559 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.
2
J.L. Palacios / Applied Mathematics and Computation 362 (2019) 124559
where x∗ is the minimal element of Sn−1 , the subset of
x ∈ R|E | : x1 ≥ x2 ≥ · · · ≥ x|E | ;
n−1 =
|E |
xj = n − 1 ,
j=1
consisting of those |E|-tuples such that 1 ≥ xi ≥ 2n for 1 ≤ i ≤ |E|. Unfortunately, we used the maximal x∗ instead of the minimal element, and thus all upper bounds for ABC(G) in [6], except those for trees, are not justified. If we consider the correct minimal element
x∗ =
n − 1 n − 1 n−1 , ,..., , |E | |E | |E |
we obtain that
ABC (G ) ≤
(2 − 1 )|E |(n − 1 ) . 2
From the fact that R−1 (G ) ≡
ABC (G ) ≤
|E | 1 (i, j )∈E di d j ≥ 2 , we can see that the bound so obtained is always worse than
(n − 1 )(|E | − R−1 (G )),
(2)
that was found in [1]. Thus, majorization on the effective resistances does not lead to better results in this context. 2. A different majorization Instead of looking at effective resistances, now we look at the quantities defined by
Ai j =
1 1 + , di dj
for all (i, j) ∈ E. It is clear that we can write
ABC (G ) ≤
2
Ai j −
2
(i, j )∈E
.
(3)
Now we can prove the following general upper bound: Proposition 1. For any n-vertex graph we have
ABC (G ) ≤ |E |
n
|E |
−
2
2
.
(4)
The equality is attained by any d-regular graph. Proof. The majorization of the summation in the right hand side of (3) is carried out by looking at the Schur-concave function
(x ) =
|E |
xi −
i=1
2
2
,
|E | on the set n of |E|-tuples x = (x1 , x2 , . . . , x|E | ) such that i=1 xi = n. Note that the Aij s belong to this set because of the general relation (i, j )∈E (diα + dαj ) = ni=1 diα +1 , which turns out the desired result when we take α = −1. Now we have that the minimal element of n is x∗ = ( |En| , |En| , . . . , |En| ), and thus
ABC (G ) ≤ ({Ai j } ) ≤ (x∗ ) = |E |
n
|E |
−
2
2
.
It is a simple calculation to verify that for any d-regular graph G the value of the bound and ABC(G) are both equal to n
d−1 2
Remarks 1. Bound (4) expressed only in terms of the number of vertices, the number of edges and the largest degree in the graph, seems to be new. It is clear that (4) and (2) are not comparable, by looking at those graphs where the equalities are attained. It should be remarked that (2) was derived without using majorization, although a similar majorization of the Aij s was used in [1] to tackle R−1 (G ), the Randic´ index, in bound (2). In recent literature some bounds similar to (4) can be found; specifically in [3] one finds that
+δ , δ
ABC (G ) < |E | √
(5)
J.L. Palacios / Applied Mathematics and Computation 362 (2019) 124559
and
ABC (G ) ≤ |E |
√ 2 − 2
δ
3
.
(6)
+δ ≥ 2, whereas It is clear that (4) is always better than (5) because by the arithmetic mean-geometric mean inequality √ δ
√ n 2 n n − < ≤ ≤ 2 , for n ≥ 2. n−1 |E | |E | 2
Also, if δ = , that is, if the graph is regular, (4) and (6) coincide, but for all other cases, (6) is weaker than (4). Indeed, for non-regular graphs we have
δ + 1 ≤ , implying
(δ + 1 )2 ≤ 3 < 3 + δ 2 , from which we get
δ 2 − δ 2 < 3 − 2 . Multiplying this inequality by δ 222 we obtain
2
δ
−
2
<
2
2 − 2
δ2
,
and taking square roots we get
2
δ
−
2
2
<
√ 2 − 2
δ
.
Finally, since 2|E| ≥ nδ , we have
2 n − ≤ | E | 2
2
δ
−
√
2
<
2
2 − 2
δ
.
We can also specify the number of pendant vertices, h, for finer results. Let us denote by E1 the set of pendant edges, that is, those edges one of whose end vertices has degree 1, and let us denote E2 = E − E1 . Also, if there are h pendant vertices, it is clear that dn−h is the smallest degree strictly larger than 1. With these notations we have the following: Proposition 2. For any n-vertex graph G with h pendant vertices we have
1
ABC (G ) ≤ h 1 +
dn−h
−
2
1
+ ( |E | − h )
n − h − h1 2 − 2, |E | − h 2
(7)
where i is the largest degree in Ei , i = 1, 2. The equality is attained by all d-regular graphs and all (d, 1)-semiregular graphs. Proof. It is clear that we can write
ABC (G ) =
(i, j )∈E
di + d j − 2 ≤ di d j
Ai j −
(i, j )∈E1
2
1
+
(i, j )∈E2
Ai j −
2
22
.
(8)
Now we will apply majorization separately on each summand. For the first summand, we have |E1 | = h, Ai j =
and dn−h ≤ d j ≤ 1 . We set (i, j )∈E1 Ai j = X, and consider the Schur-concave function 1 (x ) = |E1 | set of |E1 |-tuples such that i=1 xi = X. 1+
1 dj
The minimal element in this context is x∗ =
Ai j −
(i, j )∈E1
2
1
h 1+
1
1
2 X − . h 1
≤ (x∗ ) = h
Also, we can bound X as follows:
≤X ≤h 1+
1
X
h
, Xh , . . .
X h
|E1 | i=1
1 di
+
1 dj
=
xi − 2 on the 1
and we bound the first summand as follows:
(9)
,
(10)
2 1 − . dn−h 1
(11)
dn−h
and if we use (10) in (9), we get
(i, j )∈E1
Ai j −
2
1
≤h 1+
4
J.L. Palacios / Applied Mathematics and Computation 362 (2019) 124559
For the second summand in (8) we have |E2 | = |E | − h and Ai j = d1 + d1 with dn−h ≤ di , d j ≤ 2 . We set (i, j )∈E1 Ai j = n − X, i j
| E 2 | | E 2 | and consider the Schur-concave function 2 (x ) = i=1 xi − 22 on the set of |E2 |-tuples such that x = n − X. The i=1 i minimal element now is x∗ =
Ai j −
(i, j )∈E2
n−X n−X n−X |E |−h , |E |−h , . . . |E |−h
2
22
≤ (x∗ ) = (|E | − h )
2
and we bound the second summand as
n−X
|E | − h
−
2
22
.
(12)
Also, using (10), we can bound n − X as follows:
n−h−
h h ≤n−X ≤n−h− , dn−h 1
(13)
and if we insert (13) into (12), we get
Ai j −
(i, j )∈E2
2
22
≤ ( |E | − h )
n − h − h1 2 − 2. |E | − h 2
(14)
Inserting now (11) and (14) into (8) finishes the proof of (7). As for the equality in the case of (d, 1)-semiregular graphs, it is easy to see that in that case both (1) and (7) yield the value
1 h 1 − + ( |E | − h ) d
2 2 − 2, d d
once we notice that the sum of all degrees is 2|E | = (n − h )d + h
Remarks 2. Das found in [2] the following upper bound:
ABC (G ) ≤ h 1 −
1
+ ( |E | − h )
2 2 − 2 , dn−h dn−h
(15)
which resembles our bound (7) and also attains the equality for d-regular and (d, 1)-semiregular graphs. These bounds (15) and (7) are not comparable. Indeed, consider an n-cycle Cn to which we add n − 3 extra edges between a fixed vertex i in the cycle and all others to which it was not previously linked; now attach to every one of the vertices a copy of a 4-star graph through one of its leaves. The resulting graph Gn satisfies |V | = 4n, |E | = 5n − 3, h = 2n, d|V |−h = 1 = 3 and = 2 = di = n. Bound (15) produces
1 ABC (Gn ) ≤ 2n 1 − + (3n − 3 ) n whereas bound (7) gives
ABC (Gn ) ≤ 2n
2 + ( 3n − 3 ) 3
2 2 − ∼ 4n, 3 9
2n − 23n 2 − 2 ∼ 3.63n. 3n − 3 n
On the other hand, if we consider two n-cycles which are attached through a single vertex, this is a graph Hn satisfying
|V | = 2n − 1, |E | = 2n, h = 0, d|V |−h = 2, = 2 = 4. Bound (15) produces 2n ABC (Hn ) ≤ √ ∼ 1.41n, 2 whereas bound (7) turns out
ABC (Hn ) ≤ 2n
2 2n − 1 − ∼ 1.87n. 2n 16
References [1] [2] [3] [4]
M. Bianchi, A. Cornaro, J.L. Palacios, A. Torriero, New upper bounds for the ABC index, MATCH Commun. Math. Comput. Chem. 76 (2016) 117–130. K.C. Das, Atom-bond connectivity index of graphs, Discr. Appl. Math. 158 (2010) 1181–1188. K.C. Das, S. Elumalai, I. Gutman, On ABC index of graphs, MATCH Commun. Math. Comput. Chem. 78 (2017) 459–468. E. Estrada, L. Torres, L. Rodríguez, I. Gutman, An atom-bond connectivity index: modelling the enthalpy of formation of alkanes, Indian J. Chem. 37A (1998) 849–855. [5] E. Estrada, Atom-bond connectivity and the energetic of branched alkanes, Chem. Phys. Lett. 463 (2008) 422–425. [6] J.L. Palacios, Bounds for the augmented Zagreb and the atom-bond connectivity indices, Appl. Math. Comput. 307 (2017) 141–145. [7] R.J. Wilson, Introduction to Graph Theory, Oliver & Boyd, Edinburgh, 1972.