The maximum atom-bond connectivity index for graphs with edge-connectivity one

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The maximum atom-bond connectivity index for graphs with edge-connectivity one Qing Cui, Qiuping Qian, Lingping Zhong ∗ Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China

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Article history: Received 11 October 2016 Received in revised form 23 November 2016 Accepted 13 December 2016 Available online xxxx Keywords: Atom-bond connectivity index Edge-connectivity

The  atom-bond connectivity (ABC ) index of a graph G is defined as the sum of the weights d(u)+d(v)−2 d(u)d(v)

of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In

this short note, we determine the maximum ABC index for graphs with edge-connectivity one and characterize the corresponding extremal graphs. This answers a recent question proposed by Zhang et al. (2016). © 2016 Elsevier B.V. All rights reserved.

1. Introduction Molecular descriptors are playing a significant role in chemistry, pharmacology, etc. Among them, topological indices have a prominent place. Topological indices are numbers associated with chemical structures derived from their hydrogendepleted graphs as a tool for compact and effective description of structural formulas which are used to study and predict the structure–property correlations of organic compounds. There are lots of topological indices which have found some applications in theoretical chemistry, especially in QSPR/QSAR studies (for example, see [20, pages 34–95] and [21, pages 105–140]). Let G be a simple graph with vertex set V (G) and edge set E (G). The atom-bond connectivity (ABC ) index was introduced by Estrada et al. [13] in 1998. This index is defined as

 ABC (G) =



d(u) + d(v) − 2

uv∈E (G)

d(u)d(v)

,

where d(u) denotes the degree of a vertex u of G. It displays an excellent correlation with the heat of formation of alkanes [13], and a basically topological approach was developed on the basis of the ABC index to explain the differences in the energy of linear and branched alkanes both qualitatively and quantitatively [12]. The mathematical properties of the ABC index have been studied extensively. Furtula, Graovac and Vukičević [14] determined the maximum ABC index for trees and the maximum and minimum ABC indices for chemical trees. Xing, Zhou and Du [23] found the maximum ABC indices for trees with a perfect matching or given maximum degree, and characterized the corresponding extremal trees. See [1–11,15–19,22,25] for more information of this index. Recently, Zhang et al. [24] considered the maximum ABC indices for connected graphs with some given graph parameters such as independence number, number of pendent vertices, edge-connectivity and chromatic number. Particularly, they

∗

Corresponding author. E-mail addresses: [email protected] (Q. Cui), [email protected] (Q. Qian), [email protected] (L. Zhong).

http://dx.doi.org/10.1016/j.dam.2016.12.011 0166-218X/© 2016 Elsevier B.V. All rights reserved.

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Table 1 The ABC indices of Kn∗ and all possible graphs G∗ (for 6 ≤ n ≤ 12). n 6 7 8 9

10

n1

n2

ABC (G∗ )

ABC (Kn∗ )

n

3 3 3 4 3 4 3 4 5

3 4 5 4 6 5 7 6 5

4.9093 6.7033 8.7944 8.4854 11.1404 10.5688 13.7148 12.9093 12.6470

6.9350 9.3083 11.9021

11

14.7004

12

n1

n2

ABC (G∗ )

3 4 5 3 4 5 6

8 7 6 9 8 7 6

16.4987 15.4797 14.9839 19.4772 18.2605 17.5515 17.3180

ABC (Kn∗ ) 20.8602

24.2014

17.6901

determined the maximum ABC index for graphs with edge-connectivity k (k ≥ 2) and characterized the corresponding extremal graphs. It was suggested in [24] that the extremal graphs with the maximum ABC index for graphs with edgeconnectivity 1 might have the similar structure as in the general case (i.e., k ≥ 2). In this short note, we give an affirmative answer to this question. 2. Main result In this section, we prove the main result of this paper. First, we list the following two lemmas which will be used in our proof. The first lemma was proved by Chen and Guo [2] (see also [4]). Lemma 2.1. Let G be a graph with n vertices. If x and y are two nonadjacent vertices in G, then ABC (G) ≤ ABC (G + xy) with equality if and only if both x and y are isolated vertices of G. As a consequence, we have ABC (G) ≤ ABC (Kn ) with equality if and only if G ∼ = Kn , where Kn denotes the complete graph with n vertices. The second lemma was shown in [23,22]. Lemma 2.2. Let f (x, y) =



x+y−2 xy

with x, y ≥ 1, then f (x, y) is decreasing with respect to x for any fixed y ≥ 2.

For n ≥ 3, let Kn∗ be the graph on n vertices obtained by attaching a pendent vertex to exactly one vertex of Kn−1 . It is easy to see that Kn∗ has edge-connectivity 1. We now prove the main result of this paper by applying the similar idea as in [24]. Theorem 2.3. Let G be a graph with n ≥ 3 vertices and edge-connectivity 1. Then ABC (G) ≤



n − 3√ + 2n − 6 + (n − 2) n−1 2 n−2



2n − 5

(n − 1)(n − 2)

with equality if and only if G ∼ = Kn∗ . Proof. Suppose G∗ is the graph with the maximum ABC index among all graphs with n ≥ 3 vertices and edge-connectivity 1. First, suppose G∗ contains a vertex of degree 1, say v . Let G′ := G∗ − {v}. Then by Lemma 2.1 and by the assumption of G∗ , we see that G′ is the complete graph on n − 1 vertices; for otherwise, there must exist two nonadjacent vertices x, y in G′ such that the graph G∗ + {xy} has edge-connectivity 1 and ABC (G∗ + {xy}) > ABC (G∗ ), a contradiction. This implies that G∗ ∼ = Kn∗ , and hence the assertion holds. So we may assume that every vertex in G∗ has degree at least 2. Let e be a cut-edge in G∗ . Then G∗ − {e} has exactly two components, say G1 and G2 . By Lemma 2.1 and by the assumption of G∗ , we have both G1 and G2 as complete graphs. Let ni be the number of vertices in Gi (for i = 1, 2), then n = n1 + n2 . Without loss of generality, we may assume n2 ≥ n1 . Since G∗ has minimum degree at least 2, we have n2 ≥ n1 ≥ 3 and hence n ≥ 6. For each 6 ≤ n ≤ 12, there are only a few possible graphs satisfying the conditions that both components G1 and G2 are complete graphs and n2 ≥ n1 ≥ 3. So it is easy to check that the assertion holds for 6 ≤ n ≤ 12 through direct calculations (see Table 1). Therefore we may further assume that n ≥ 13. Since n2 ≥ n1 ≥ 3 (and n ≥ 13), by Lemma 2.2, we have ABC (Kn∗ ) = f (1, n − 1) + (n − 2)f (n − 1, n − 2) +

≥ f (1, n − 1) + (n − 2)f (n − 1, n − 1) +

(n − 2)(n − 3) 2

(n − 2)(n − 3)

(n − 1)(n − 2) = f (1, n − 1) + f (n − 1, n − 1) 2  3 n−2 (n − 2) 2 = + √ n−1 2

2

f (n − 2, n − 2) f (n − 1, n − 1)

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3

and

(n1 − 1)(n1 − 2)

ABC (G∗ ) = (n1 − 1)f (n1 , n1 − 1) +

f (n1 − 1, n1 − 1) 2 (n2 − 1)(n2 − 2) f (n2 − 1, n2 − 1) + f (n1 , n2 ) + (n2 − 1)f (n2 , n2 − 1) + 2 (n1 − 1)(n1 − 2) ≤ (n1 − 1)f (n1 − 1, n1 − 1) + f (n1 − 1, n1 − 1) 2 (n2 − 1)(n2 − 2) f (n2 − 1, n2 − 1) + f (n1 , n2 ) + (n2 − 1)f (n2 − 1, n2 − 1) + 2 n2 (n2 − 1) n1 (n1 − 1) f (n1 − 1, n1 − 1) + f (n2 − 1, n2 − 1) + f (n1 , n2 ) = 2 2 n1  n2  = √ n1 − 2 + √ n2 − 2 + 2 2 n

3 2

n

3 2



< √1 + √2 + 2

n1 + n2 − 2 n1 n2

2



n1 + n2 − 2 n1 n2

.

By the assumption of G∗ , to derive a contradiction, it suffices to show the inequality



n−2 n−1

3

3

2

2

holds for n ≥ 13 and 3 ≤ n1 ≤



3

n2 n2 (n − 2) 2 > √1 + √2 + + √ n . 2

3 2

3 2



n1 + n2 − 2

(∗)

n1 n2

2

Note that (∗) is equivalent to

(n − 2) − n1 − (n − n1 )

3 2

 +



2(n − 2)





1

−

n−1



1

>0

n1 (n − n1 )

since n = n1 + n2 . Let h(n, n1 ) =



3 2

3 2

(n − 2) − n1 − (n − n1 )

with n ≥ 13 and 3 ≤ n1 ≤

n . 2

3 2

 +



2(n − 2)





1 n−1

−



1 n1 (n − n1 )

Since

 √  3 √ n−2 ∂ h(n, n1 ) = n − n1 − n1 + (n − 2n1 ) ≥ 0, 3 ∂ n1 2 2n1 (n − n1 )3 we know that h(n, n1 ) is increasing with respect to n1 for n ≥ 13 and 3 ≤ n1 ≤

   3 3 3 h(n, n1 ) ≥ h(n, 3) = (n − 2) 2 − 3 2 − (n − 3) 2 + 2(n − 2)



n . 2

1 n−1

Hence

 −

1 3(n − 3)

 .

We now show h(n, 3) > 0 for all n ≥ 13, which implies (∗) holds. For each 13 ≤ n ≤ 20, it is easy to check that h(n, 3) > 0 through direct calculations. If n ≥ 21, then we have 3

3

3n2 − 15n + 19

3

(n − 2) 2 − 3 2 − (n − 3) 2 = √

n3 − 6n2 + 12n − 8 +

3n2 − 15n

> √

n3 +

√

√

3

− 32

n3 15 1

√ 3√ n− √ −3 3 2 2 n √ 3√ 15 1 ≥ 21 − √ −3 3 2 2 21 >0 =

n3 − 9n2 + 27n − 27

3

− 32

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and



2(n − 2)



1 n−1

 −

1 3(n − 3)

 > 0.

Hence h(n, 3) > 0 for n ≥ 21. This completes the proof of the theorem.



Acknowledgments The authors would like to thank the anonymous referees for their helpful comments and suggestions which improved the paper. This work was supported by the National Natural Science Foundation of China (No. 11501291), the Fundamental Research Funds for the Central Universities (Nos. NS2015078 and NZ2015108) and China Scholarship Council. References [1] M.B. Ahmadi, D. Dimitrov, I. Gutman, S.A. Hosseini, Disproving a conjecture on trees with minimal atom-bond connectivity index, MATCH Commun. Math. Comput. Chem. 72 (2014) 685–698. [2] J. Chen, X. Guo, Extreme atom-bond connectivity index of graphs, MATCH Commun. Math. Comput. Chem. 65 (2011) 713–722. [3] K.C. Das, Atom-bond connectivity index of graphs, Discrete Appl. Math. 158 (2010) 1181–1188. [4] K.C. Das, I. Gutman, B. Furtula, On atom-bond connectivity index, Chem. Phys. Lett. 511 (2011) 452–454. [5] K.C. Das, I. Gutman, B. Furtula, On atom-bond connectivity index, Filomat 26 (2012) 733–738. [6] K.C. Das, M.A. Mohammed, I. Gutman, K.A. Atan, Comparison between atom-bond connectivity indices of graphs, MATCH Commun. Math. Comput. Chem. 76 (2016) 159–170. [7] K.C. Das, N. Trinajstić, Comparison between first geometric-arithmetic index and atom-bond connectivity index, Chem. Phys. Lett. 497 (2010) 149–151. [8] D. Dimitrov, On structural properties of trees with minimal atom-bond connectivity index, Discrete Appl. Math. 172 (2014) 28–44. [9] D. Dimitrov, On structural properties of trees with minimal atom-bond connectivity index II: Bounds on B1 - and B2 -branches, Discrete Appl. Math. 204 (2016) 90–116. [10] D. Dimitrov, Z. Du, C.M. da Fonseca, On structural properties of trees with minimal atom-bond connectivity index III: Trees with pendent paths of length three, Appl. Math. Comput. 282 (2016) 276–290. [11] Z. Du, C.M. da Fonseca, On a family of trees with minimal atom-bond connectivity index, Discrete Appl. Math. 202 (2016) 37–49. [12] E. Estrada, Atom-bond connectivity and the energetic of branched alkanes, Chem. Phys. Lett. 463 (2008) 422–425. [13] 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. [14] B. Furtula, A. Graovac, D. Vukičević, Atom-bond connectivity index of trees, Discrete Appl. Math. 157 (2009) 2828–2835. [15] Y. Gao, Y. Shao, The smallest ABC index of trees with n pendent vertices, MATCH Commun. Math. Comput. Chem. 76 (2016) 141–158. [16] M. Goubko, C. Magnant, P.S. Nowbandegani, I. Gutman, ABC index of trees with fixed number of leaves, MATCH Commun. Math. Comput. Chem. 74 (2015) 697–702. [17] S.A. Hosseini, M.B. Ahmadi, I. Gutman, Kragujevac trees with minimal atom-bond connectivity index, MATCH Commun. Math. Comput. Chem. 71 (2014) 5–20. [18] W. Lin, J. Chen, C. Ma, Y. Zhang, J. Chen, D. Zhang, F. Jia, On trees with minimal ABC index among trees with given number of leaves, MATCH Commun. Math. Comput. Chem. 76 (2016) 131–140. [19] C. Magnant, P.S. Nowbandegani, I. Gutman, Which tree has the smallest ABC index among trees with k leaves, Discrete Appl. Math. 194 (2015) 143–146. [20] R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000. [21] N. Trinajstić, Chemical Graph Theory, Vol. II, CRC Press, Boca Raton, Florida, 1983. [22] R. Xing, B. Zhou, F. Dong, On atom-bond connectivity index of connected graphs, Discrete Appl. Math. 159 (2011) 1617–1630. [23] R. Xing, B. Zhou, Z. Du, Further results on atom-bond connectivity index of trees, Discrete Appl. Math. 158 (2010) 1536–1545. [24] X.M. Zhang, Y. Yang, H. Wang, X.D. Zhang, Maximum atom-bond connectivity index with given graph parameters, Discrete Appl. Math. 215 (2016) 208–217. [25] L. Zhong, Q. Cui, On a relation between the atom-bond connectivity and the first geometric-arithmetic indices, Discrete Appl. Math. 185 (2015) 249–253.