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Some lower and upper bounds on the third ABC index
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ScienceDirect AKCE International Journal of Graphs and Combinatorics 13 (2016) 11–15 www.elsevier.com/locate/akcej
Some lower and upper bounds on the third ABC index Dae-Won Lee Department of Chemical Engineering, Sungkyunkwan University, Suwon 440-746, Republic of Korea Received 8 April 2014; accepted 16 February 2016 Available online 9 March 2016
Abstract Atom-bond connectivity (ABC) index has been applied up to now to study the stability of alkanes and the strain energy of cycloalkanes. Graovac defined the second ABC index as
ABC2 (G) =
vi v j ∈E(G)
1 1 2 + − , ni nj ni n j
and Kinkar studied the upper bounds. In this paper, we define a new index which is called the third ABC index and it is defined as
ABC3 (G) =
vi v j ∈E(G)
1 1 2 + − , ei ej ei e j
(1)
and we present some lower and upper bounds on ABC3 index of graphs. c 2016 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Molecular graph; Atom-bond connectivity (ABC) index; The third Atom-bond connectivity (ABC3 ) index; Topological indices
1. Introduction Molecular descriptors are playing significant role in chemistry, pharmacology, etc. Among them, topological indices have a prominent place [1]. One of the best known and widely used index is the connectivity index, χ , introduced in 1975 by Milan Randi´c [2], who has shown that this index reflects molecular branching. Some novel results about branching can be found in [3–7] and in the references cited therein. However, many physico-chemical properties are dependent on factors other than branching. The lower and upper bounds on ABC index of chemical trees in terms of the number of vertices were obtained in [8]. Also, it has been shown that the star K 1,n−1 , has the maximal ABC value of trees.
Peer review under responsibility of Kalasalingam University. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.akcej.2016.02.002 c 2016 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 0972-8600/⃝ (http://creativecommons.org/licenses/by-nc-nd/4.0/).
12
D.-W. Lee / AKCE International Journal of Graphs and Combinatorics 13 (2016) 11–15
Let G = (V, E) be a simple connected graph with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G), where |V (G)| = n and |E(G)| = m. Let x, y ∈ V (G), then the distance dG (x, y) between x and y is defined as the length of any shortest path in G connecting x and y. For a vertex vi of V (G), its eccentricity ei is the largest distance between vi and any other vertex vk of G, i.e, ei = maxv j ∈V (G) dG (vi , v j ). The diameter d(G) of G is defined as the maximum eccentricity of G. Similarly, the radius r (G) is defined as the minimum eccentricity of G. There are some topological indices which are related to the eccentricity like the eccentric connectivity index di ei ξ(G) = vi ∈V (G)
where di is the degree of the vertex vi . The Zagreb indices have been introduced more than 30 years ago by Gutman and Trinajsti´c [9]. They are defined as M1 (G) = di2 , vi ∈V (G)
M2 (G) =
di d j .
vi v j ∈E(G)
(see [10–13] and the references therein). Also, there are similar indices called Zagreb eccentricity indices which are also related to eccentricity. Zagreb eccentricity indices are defined as E 1 (G) = ei2 , vi ∈V (G)
E 2 (G) =
ei e j ,
vi v j ∈E(G)
(see [14–17] and the references therein). In order to take this into account but at the same time to keep the spirit of the Randi´c index, Estrada et al. proposed a new index, nowadays known as the atom-bond connectivity (ABC) index, which is defined as [18] 1 1 2 ABC(G) = + − . di dj di d j v v ∈E(G) i j
The ABC index has proven to be a valuable predictive index in the study of the heat formation of alkanes [18,19], and some mathematical properties are obtained in [20,21]. Very recently, Graovac [22] defined the second ABC index as 1 1 2 ABC2 (G) = + − , ni nj ni n j v v ∈E(G) i j
which was given by replacing the vertex degree to n i , where n i is the number of vertices of G whose distance to the vertex vi is smaller than the distance to the vertex v j . Also, Kinkar establised some mathematical properties of the second ABC index [23]. Now we define the third ABC index; ABC3 (G), 1 1 2 ABC3 (G) = + − (2) ei ej ei e j v v ∈E(G) i j
which was given by replacing the vertex degree to eccentricity, ei . In this paper, we present lower and upper bounds on ABC3 index of connected, simple graphs in terms of other indices; the first Zagreb index, the second Zagreb index, eccentric connectivity index, the first Zagreb eccentricity index, and the second Zagreb eccentricity index.
D.-W. Lee / AKCE International Journal of Graphs and Combinatorics 13 (2016) 11–15
13
2. Lower and upper bounds on ABC3 index √ √ , ABC3 (C n ) = 2 n − 2 if n is even, By direct calculation, we can find ABC3 (K n ) = 0, ABC3 (K 1,n−1 ) = n−1 2 √ 2n ABC3 (Cn ) = n−1 n − 3 if n is odd, where K n , K 1,n−1 and Cn denote the complete graph, star graph and cycle with n vertices, respectively. We can easily see that the complete graph K n gives the minimal value for ABC3 index, so in this paper we only consider when graph G differs from K n . Theorem 2.1. Let G be a simple connected graph. Then ABC3 (G) ≥ √
1 E 2 (G)
(3)
where E 2 (G) is the second Zagreb eccentricity index. Proof. Since G K n , it is easy to see that for every e = vi v j in E(G), ei + e j ≥ 3. By definition of ABC3 index, we have 1 1 2 1 1 1 ABC3 (G) = . + − ≥ ≥ =√ √ e e e e e e E i j i j i j 2 (G) ei e j vi v j ∈E(G) vi v j ∈E(G) vi v j ∈E(G)
Theorem 2.2. Let G be a simple connected graph with m edges, radius r = r (G) ≥ 2, diameter d = d(G). √ √ 2m √ 2m √ d − 1 ≤ ABC3 (G) ≤ r − 1. d r Equality holds if and only if G is a self-centered graph. Proof. For 2 ≤ r ≤ ei , e j ≤ d, we have 1 2 1 1 2 2 1 + − ≥ + 1− as e j ≤ d and 1 − ≥0 ei ej ei e j ei d ei ei 1 1 2 = + 1− d ei d 2 2 ≥ 2 (d − 1), as ei ≤ d and 1 − ≥ 0, d d with equality holding if and only if ei = e j = d. Similarly, we can easily show that 1 2 2 1 + − ≤ 2 (r − 1), ei ej ei e j r
for 2 ≤ r ≤ ei , e j ≤ d
with equality holding if and only if ei = e j = r .
(4)
(5)
(6)
Lemma 2.3 ([24]). Let (a1 , a2 , . . . , an ) be a positive n-tuple such that there exist positive numbers A, a satisfying: 0 < a ≤ ai ≤ A. Then n
n i=1
n
ai2
ai
1 2 ≤ 4
A + a
2 a . A
(7)
i=1
Equality holds if and only if a = A or q=
A/a n A/a + 1
is an integer and q of the numbers ai coincide with a and the remaining n − q of the ai ’s coincide with A(̸=a).
(8)
14
D.-W. Lee / AKCE International Journal of Graphs and Combinatorics 13 (2016) 11–15
Theorem 2.4. Let G be a simple connected graph with m edges, radius r = r (G) ≥ 2, diameter d = d(G). Then √ 4m (r − 1)(d − 1) ABC3 (G) ≥ . (9) 2 1√ √ 1 r d r r − 1 + d d − 1 E 2 (G) Proof. We know that √ √ 2√ 1 2√ 1 2 d −1≤ r − 1. + − ≤ d ei ej ei e j r Also by Lemma 2.3, we have 2 1 1 2 + − ≥ ei ej ei e j v v ∈E(G) i j
√ 1 1 2 4m (r − 1)(d − 1) + − . √ 2 × √ ei ej ei e j vi v j ∈E(G) r d r1 r − 1 + d1 d − 1
Since ei + e j ≥ 3 1 1 2 1 + − ≥ ≥ ei ej ei e j ee v v ∈E(G) v v ∈E(G) i j i j
i j
Hence it follows that ABC3 (G) ≥
(10)
1 vi v j ∈E(G)
√ 4m (r − 1)(d − 1) . √ 2 √ 1 1 r d r r − 1 + d d − 1 E 2 (G)
ei e j
=
1 . E 2 (G)
(11)
(12)
(13)
Theorem 2.5. Let G be a simple connected graph with n vertices and m edges. Then 1
1 ≤ ABC3 (G) ≤ √ 2nm 2 − m M1 (G) − 2m 2 2 n 2 m − n M1 (G) + M2 (G)
(14)
where M1 (G) and M2 (G) are the first Zagreb index and the second Zagreb index, respectively. Proof. From the proof of Theorem 2.1, we already have 1 1 2 1 + − ≥ . ei ej ei e j ei e j v v ∈E(G) i j
(15)
vi v j ∈E(G)
Since ei ≤ n − di , we get 1
vi v j ∈E(G)
ei e j
≥
1 n2m
− n M1 (G) + M2 (G)
.
This completes the lower bound in (14). Now, since G K n , ei e j ≥ 2 for vi v j ∈ E(G), we have 1 1 2 1 + − ≤√ ei + e j − 2. ei ej ei e j 2 v v ∈E(G) v v ∈E(G) i j
i j
(16)
(17)
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D.-W. Lee / AKCE International Journal of Graphs and Combinatorics 13 (2016) 11–15
By Cauchy–Schwarz inequality, 1 1 ei + e j − 2 ≤ √ 1· (ei + e j − 2) √ 2 vi v j ∈E(G) 2 vi v j ∈E(G) vi v j ∈E(G)
(18)
since ei ≤ n − di for every vi ∈ V (G), we get 1 1 (ei + e j − 2) ≤ √ 2nm 2 − m M1 (G) − 2m 2 . 1· √ 2 vi v j ∈E(G) vi v j ∈E(G) 2
(19)
Theorem 2.6. Let G be a simple connected graph with n vertices, m edges, radius r, and eccentric connectivity index ξ(G). Then √ 1 2 1 m + − ≤ (ξ(G) − 2m). (20) ei ej ei e j r v v ∈E(G) i j
Proof. By Cauchy–Schwarz inequality, we have 1 1 1 2 (ei + e j − 2) + − ≤ . ei ej ei e j ee v v ∈E(G) v v ∈E(G) v v ∈E(G) i j i j
Also,
vi v j ∈E(G) (ei
i j
(21)
i j
+ e j − 2) = ξ(G) − 2m, ei ≥ r for all vi ∈ V (G). Hence the theorem follows.
Acknowledgments The author would like to thank anonymous referees very much for valuable suggestions, and discussions which result in a great improvement. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000. M. Randi´c, On characterization of molecular branching, J. Am. Chem. Soc. 97 (1975) 6609–6615. ˇ I. Gutman, D. Vukiˇcevi´c, J. Zerovnik, A class of modified Wiener indices, Croat. Chem. Acta. 77 (2004) 103–109. D. Vukiˇcevi´c, Distinction between modifications of Wiener indices, MATCH Commun. Math. Comput. Chem. 47 (2003) 87–105. D. Vukiˇcevi´c, I. Gutman, Note on a class of modified Wiener indices, MATCH Commun. Math. Comput. Chem. 47 (2003) 107–117. ˇ D. Vukiˇcevi´c, J. Zerovnik, Variable Wiener indices, MATCH Commun. Math. Comput. Chem. 53 (2005) 385–402. ˇ D. Vukiˇcevi´c, J. Zerovnik, New indices based on the modified Wiener indices, MATCH Commun. Math. Comput. Chem. 60 (2008) 119–132. B. Furtula, A. Graovac, D. Vukiˇcevi´c, Atom-bond connectivity index of trees, Discrete Appl. Math. 157 (2009) 2828–2835. I. Gutman, N. Trinajsti´c, Graph theory and molecular orbitals. III. Total-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538. K.C. Das, Maximizing the sum of the squares of the degrees of a graph, Discrete Math. 285 (2004) 57–66. K.C. Das, I. Gutman, Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem. 52 (2004) 103–112. K.C. Das, I. Gutman, B. Zhou, New upper bounds on Zagreb indices, J. Math. Chem. 46 (2009) 514–521. I. Gutman, K.C. Das, The first Zagreb indices 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83–92. K.C. Das, D.-W. Lee, A. Graovac, Some properties of the Zagreb eccentricity indices, Ars Math. Contemp. 6 (2013) 117–125. D.-W. Lee, Lower and upper bounds of Zagreb eccentricity indices on unicyclic graphs, Adv. Appl. Math. Sci. 12 (7) (2013) 403–410. M. Ghorbani, M.A. Hosseinzadeh, A new version of Zagreb indices, Filomat 26 (2012) 93–100. D. Vukiˇcevi´c, A. Graovac, Note on the comparison of the first and second normalized Zagreb eccentricity indices, Acta Chim. Slov. 57 (2010) 524–528. E. Estrada, L. Torres, L. Rodr´ıguez, I. Gutman, An atom-bond connectivity index: Modelling the enthalpy of formation of alkanes, Indian J. Chem. A 37 (1998) 849–855. E. Estrada, Atom-bond connectivity and the energetic of branched alkanes, Chem. Phys. Lett. 463 (2008) 422–425. K.C. Das, Atom-bond connectivity index of graphs, Discrete Appl. Math. 158 (2010) 1181–1188. K.C. Das, I. Gutman, B. Furtula, On atom-bond connectivity index, Chem. Phys. Lett. 511 (2011) 452–454. A. Graovac, M. Ghorbani, Acta Chim. Slov. 57 (2010) 609–612. K.C. Das, K. Xu, A. Graovac, Maximal unicyclic graphs with respect to new Atom-bond connectivity index, Acta Chim. Slov. 60 (2012) 34–42. G. P´olya, G. Szeg¨o, Probelms and theorems in analysis, in: Series, Integral Calculus, Theory of Functions, vol. I, Springer-Verlag, 1972.
ScienceDirect AKCE International Journal of Graphs and Combinatorics 13 (2016) 11–15 www.elsevier.com/locate/akcej
Some lower and upper bounds on the third ABC index Dae-Won Lee Department of Chemical Engineering, Sungkyunkwan University, Suwon 440-746, Republic of Korea Received 8 April 2014; accepted 16 February 2016 Available online 9 March 2016
Abstract Atom-bond connectivity (ABC) index has been applied up to now to study the stability of alkanes and the strain energy of cycloalkanes. Graovac defined the second ABC index as
ABC2 (G) =
vi v j ∈E(G)
1 1 2 + − , ni nj ni n j
and Kinkar studied the upper bounds. In this paper, we define a new index which is called the third ABC index and it is defined as
ABC3 (G) =
vi v j ∈E(G)
1 1 2 + − , ei ej ei e j
(1)
and we present some lower and upper bounds on ABC3 index of graphs. c 2016 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Molecular graph; Atom-bond connectivity (ABC) index; The third Atom-bond connectivity (ABC3 ) index; Topological indices
1. Introduction Molecular descriptors are playing significant role in chemistry, pharmacology, etc. Among them, topological indices have a prominent place [1]. One of the best known and widely used index is the connectivity index, χ , introduced in 1975 by Milan Randi´c [2], who has shown that this index reflects molecular branching. Some novel results about branching can be found in [3–7] and in the references cited therein. However, many physico-chemical properties are dependent on factors other than branching. The lower and upper bounds on ABC index of chemical trees in terms of the number of vertices were obtained in [8]. Also, it has been shown that the star K 1,n−1 , has the maximal ABC value of trees.
Peer review under responsibility of Kalasalingam University. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.akcej.2016.02.002 c 2016 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 0972-8600/⃝ (http://creativecommons.org/licenses/by-nc-nd/4.0/).
12
D.-W. Lee / AKCE International Journal of Graphs and Combinatorics 13 (2016) 11–15
Let G = (V, E) be a simple connected graph with vertex set V (G) = {v1 , v2 , . . . , vn } and edge set E(G), where |V (G)| = n and |E(G)| = m. Let x, y ∈ V (G), then the distance dG (x, y) between x and y is defined as the length of any shortest path in G connecting x and y. For a vertex vi of V (G), its eccentricity ei is the largest distance between vi and any other vertex vk of G, i.e, ei = maxv j ∈V (G) dG (vi , v j ). The diameter d(G) of G is defined as the maximum eccentricity of G. Similarly, the radius r (G) is defined as the minimum eccentricity of G. There are some topological indices which are related to the eccentricity like the eccentric connectivity index di ei ξ(G) = vi ∈V (G)
where di is the degree of the vertex vi . The Zagreb indices have been introduced more than 30 years ago by Gutman and Trinajsti´c [9]. They are defined as M1 (G) = di2 , vi ∈V (G)
M2 (G) =
di d j .
vi v j ∈E(G)
(see [10–13] and the references therein). Also, there are similar indices called Zagreb eccentricity indices which are also related to eccentricity. Zagreb eccentricity indices are defined as E 1 (G) = ei2 , vi ∈V (G)
E 2 (G) =
ei e j ,
vi v j ∈E(G)
(see [14–17] and the references therein). In order to take this into account but at the same time to keep the spirit of the Randi´c index, Estrada et al. proposed a new index, nowadays known as the atom-bond connectivity (ABC) index, which is defined as [18] 1 1 2 ABC(G) = + − . di dj di d j v v ∈E(G) i j
The ABC index has proven to be a valuable predictive index in the study of the heat formation of alkanes [18,19], and some mathematical properties are obtained in [20,21]. Very recently, Graovac [22] defined the second ABC index as 1 1 2 ABC2 (G) = + − , ni nj ni n j v v ∈E(G) i j
which was given by replacing the vertex degree to n i , where n i is the number of vertices of G whose distance to the vertex vi is smaller than the distance to the vertex v j . Also, Kinkar establised some mathematical properties of the second ABC index [23]. Now we define the third ABC index; ABC3 (G), 1 1 2 ABC3 (G) = + − (2) ei ej ei e j v v ∈E(G) i j
which was given by replacing the vertex degree to eccentricity, ei . In this paper, we present lower and upper bounds on ABC3 index of connected, simple graphs in terms of other indices; the first Zagreb index, the second Zagreb index, eccentric connectivity index, the first Zagreb eccentricity index, and the second Zagreb eccentricity index.
D.-W. Lee / AKCE International Journal of Graphs and Combinatorics 13 (2016) 11–15
13
2. Lower and upper bounds on ABC3 index √ √ , ABC3 (C n ) = 2 n − 2 if n is even, By direct calculation, we can find ABC3 (K n ) = 0, ABC3 (K 1,n−1 ) = n−1 2 √ 2n ABC3 (Cn ) = n−1 n − 3 if n is odd, where K n , K 1,n−1 and Cn denote the complete graph, star graph and cycle with n vertices, respectively. We can easily see that the complete graph K n gives the minimal value for ABC3 index, so in this paper we only consider when graph G differs from K n . Theorem 2.1. Let G be a simple connected graph. Then ABC3 (G) ≥ √
1 E 2 (G)
(3)
where E 2 (G) is the second Zagreb eccentricity index. Proof. Since G K n , it is easy to see that for every e = vi v j in E(G), ei + e j ≥ 3. By definition of ABC3 index, we have 1 1 2 1 1 1 ABC3 (G) = . + − ≥ ≥ =√ √ e e e e e e E i j i j i j 2 (G) ei e j vi v j ∈E(G) vi v j ∈E(G) vi v j ∈E(G)
Theorem 2.2. Let G be a simple connected graph with m edges, radius r = r (G) ≥ 2, diameter d = d(G). √ √ 2m √ 2m √ d − 1 ≤ ABC3 (G) ≤ r − 1. d r Equality holds if and only if G is a self-centered graph. Proof. For 2 ≤ r ≤ ei , e j ≤ d, we have 1 2 1 1 2 2 1 + − ≥ + 1− as e j ≤ d and 1 − ≥0 ei ej ei e j ei d ei ei 1 1 2 = + 1− d ei d 2 2 ≥ 2 (d − 1), as ei ≤ d and 1 − ≥ 0, d d with equality holding if and only if ei = e j = d. Similarly, we can easily show that 1 2 2 1 + − ≤ 2 (r − 1), ei ej ei e j r
for 2 ≤ r ≤ ei , e j ≤ d
with equality holding if and only if ei = e j = r .
(4)
(5)
(6)
Lemma 2.3 ([24]). Let (a1 , a2 , . . . , an ) be a positive n-tuple such that there exist positive numbers A, a satisfying: 0 < a ≤ ai ≤ A. Then n
n i=1
n
ai2
ai
1 2 ≤ 4
A + a
2 a . A
(7)
i=1
Equality holds if and only if a = A or q=
A/a n A/a + 1
is an integer and q of the numbers ai coincide with a and the remaining n − q of the ai ’s coincide with A(̸=a).
(8)
14
D.-W. Lee / AKCE International Journal of Graphs and Combinatorics 13 (2016) 11–15
Theorem 2.4. Let G be a simple connected graph with m edges, radius r = r (G) ≥ 2, diameter d = d(G). Then √ 4m (r − 1)(d − 1) ABC3 (G) ≥ . (9) 2 1√ √ 1 r d r r − 1 + d d − 1 E 2 (G) Proof. We know that √ √ 2√ 1 2√ 1 2 d −1≤ r − 1. + − ≤ d ei ej ei e j r Also by Lemma 2.3, we have 2 1 1 2 + − ≥ ei ej ei e j v v ∈E(G) i j
√ 1 1 2 4m (r − 1)(d − 1) + − . √ 2 × √ ei ej ei e j vi v j ∈E(G) r d r1 r − 1 + d1 d − 1
Since ei + e j ≥ 3 1 1 2 1 + − ≥ ≥ ei ej ei e j ee v v ∈E(G) v v ∈E(G) i j i j
i j
Hence it follows that ABC3 (G) ≥
(10)
1 vi v j ∈E(G)
√ 4m (r − 1)(d − 1) . √ 2 √ 1 1 r d r r − 1 + d d − 1 E 2 (G)
ei e j
=
1 . E 2 (G)
(11)
(12)
(13)
Theorem 2.5. Let G be a simple connected graph with n vertices and m edges. Then 1
1 ≤ ABC3 (G) ≤ √ 2nm 2 − m M1 (G) − 2m 2 2 n 2 m − n M1 (G) + M2 (G)
(14)
where M1 (G) and M2 (G) are the first Zagreb index and the second Zagreb index, respectively. Proof. From the proof of Theorem 2.1, we already have 1 1 2 1 + − ≥ . ei ej ei e j ei e j v v ∈E(G) i j
(15)
vi v j ∈E(G)
Since ei ≤ n − di , we get 1
vi v j ∈E(G)
ei e j
≥
1 n2m
− n M1 (G) + M2 (G)
.
This completes the lower bound in (14). Now, since G K n , ei e j ≥ 2 for vi v j ∈ E(G), we have 1 1 2 1 + − ≤√ ei + e j − 2. ei ej ei e j 2 v v ∈E(G) v v ∈E(G) i j
i j
(16)
(17)
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D.-W. Lee / AKCE International Journal of Graphs and Combinatorics 13 (2016) 11–15
By Cauchy–Schwarz inequality, 1 1 ei + e j − 2 ≤ √ 1· (ei + e j − 2) √ 2 vi v j ∈E(G) 2 vi v j ∈E(G) vi v j ∈E(G)
(18)
since ei ≤ n − di for every vi ∈ V (G), we get 1 1 (ei + e j − 2) ≤ √ 2nm 2 − m M1 (G) − 2m 2 . 1· √ 2 vi v j ∈E(G) vi v j ∈E(G) 2
(19)
Theorem 2.6. Let G be a simple connected graph with n vertices, m edges, radius r, and eccentric connectivity index ξ(G). Then √ 1 2 1 m + − ≤ (ξ(G) − 2m). (20) ei ej ei e j r v v ∈E(G) i j
Proof. By Cauchy–Schwarz inequality, we have 1 1 1 2 (ei + e j − 2) + − ≤ . ei ej ei e j ee v v ∈E(G) v v ∈E(G) v v ∈E(G) i j i j
Also,
vi v j ∈E(G) (ei
i j
(21)
i j
+ e j − 2) = ξ(G) − 2m, ei ≥ r for all vi ∈ V (G). Hence the theorem follows.
Acknowledgments The author would like to thank anonymous referees very much for valuable suggestions, and discussions which result in a great improvement. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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