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Complete solution to a conjecture on the Randić index of triangle-free graphs
Discrete Mathematics 309 (2009) 6322–6324
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Note
Complete solution to a conjecture on the Randić index of triangle-free graphsI Xueliang Li ∗ , Jianxi Liu Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin 300071, PR China
article
info
Article history: Received 4 January 2009 Received in revised form 31 May 2009 Accepted 12 June 2009 Available online 26 June 2009 Keywords: Randić index Conjecture Triangle-free graph
abstract 1
−2 , where d(u) is The Randić index R(G) of a graph G is defined by R(G) = uv (d(u)d(v)) the degree of a vertex u in G and the summation extends over all edges uv of G. A conjecture about the Randić index says that for any √ triangle-free graph G of order n with minimum degree δ ≥ k ≥ 1, one has R(G) ≥ k(n − k), where the equality holds if and only if G = Kk,n−k . In this short note we give a confirmative proof for the conjecture. © 2009 Elsevier B.V. All rights reserved.
P
1. Introduction The Randić index of a graph G was introduced by the chemist Milan Randić in 1975, which is defined by R(G) =
X 1 (d(u)d(v))− 2 , uv
where d(u) is the degree of a vertex u in G and the summation extends over all edges uv of G. This topological index is suitable for measuring the extent of branching of the carbon-atom skeleton of saturated hydrocarbons. Randić himself demonstrated [12] that his index is well correlated with a variety of physico-chemical properties of alkanes. Nowadays the index becomes one of the most popular molecular descriptors √ to which three books are devoted [5–7]. Bollobás and Erdős [1] gave the sharp lower bound R(G) ≥ n − 1 for G being a graph of order n with minimum degree at least 1. In [3] Fajtlowitcz mentioned that Bollobás and Erdős asked for the minimum value on the Randić index among the graphs with given minimum degree δ . Delorme, Favaron and Rautenbach [2] answered this question for δ = 2 and proposed a conjecture concerning the minimum value of the index R for all graphs of order n with minimum degree δ . They also gave a best possible lower bound for the index R of a triangle-free graph with given lower bound of the minimum degree δ : Let G be a triangle-free graph of order n with minimum degree δ ≥ k ≥ 1. Then R(G) ≥
p
k(n − k),
where the equality holds if and only if G = Kk,n−k . Liu, Lu and Tian [11] pointed out a mistake in the proof of this result, and gave a new proof only for the case when k = 2. It is proposed as a conjecture by one of the present authors in [7,9,10] as follows:
I Supported by NSFC No.10831001, PCSIRT and the ‘‘973’’ program.
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (X. Li).
0012-365X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2009.06.003
X. Li, J. Liu / Discrete Mathematics 309 (2009) 6322–6324
6323
Conjecture 1.1. Let G be a triangle-free graph of order n with minimum degree δ ≥ k ≥ 1. Then R(G) ≥
p
k(n − k),
where the equality holds if and only if G = Kk,n−k . In this paper, we want to prove that Conjecture 1.1 is true, completely solving the conjecture. 2. Main result The following result was obtained by Favaron et al. in [4], and is need in what follows. Lemma 2.1 ([4]). For any triangle-free graph G with m edges, we have R(G) ≥
√
m.
Now we come to the proof of Conjecture 1.1. Theorem 2.2. For any triangle-free graph G of order n and minimum degree δ ≥ k ≥ 1, we have R(G) ≥
p
k(n − k).
Equality holds if and only if G = Kk,n−k . Proof. Let m denote the number of edges of G and let xi,j denote the number of edges uv of G with {dG (u), dG (v)} = {i, j}. If m > k(n − k), then the result follows from Lemma 2.1 and equality is not possible. Hence we may assume m ≤ k(n − k). Since G is triangle-free, adjacent vertices have disjoint neighborhoods which imply that ∆ ≤ n − δ . From Eq. (2.1.13) of [7] we know that the Randić index can be rewritten as R =
≥
=
n 2 n 2 n
−
−
2 δ≤i≤j≤∆ 1
X
2 δ≤i≤j≤n−δ 1
1
j
i
1
1
2
√ −√ n−δ δ 2 X
xi,j
1
xi,j √ −√ n−δ δ δ≤i≤j≤n−δ 2 1 n m 1 = − √ −√ 2 2 n−δ δ 2 k(n − k) 1 1 n ≥ − √ −√ 2 2 n−δ δ 2 p n k(n − k) 1 1 ≥ − = k(n − k). √ −√ 2 2 n−k k 2
−
2 X 1 1 − xi,j √ √
1
2
In the above inequality chain equality holds throughout if and only if m = k(n − k) = xk,n−k , which immediately implies that G = Kk,n−k . Remark. Li, Liu and Liu [8] completely solved the problem for the minimum value on the Randić index for the graphs with given minimum degree δ . Theorem 2.2 in the present paper completely solves the conjecture for graphs without triangles. Up until now, the conjectures proposed in [2] have been completely solved. Acknowledgements The authors would like to express thanks to two anonymous referees for detailed comments and suggestions, which helped to shorten the paper dramatically. References [1] [2] [3] [4]
B. Bollobás, P. Erdős, Graphs of extremal weights, Ars Combin. 50 (1998) 225–233. C. Delorme, O. Favaron, D. Rautenbach, On the Randić index, Discrete Math. 257 (1) (2002) 29–38. S. Fajtlowicz, Written on the wall, conjectures derived on the basis of the program Galatea Gabriella Graffiti, University of Houston, 1998. O. Favaron, M. Mahéo, J.F. Saclé, Some eigenvalue properties in graphs (Conjecture of Graffiti-II), Discrete Math. 111 (1–3) (1993) 197–220.
6324
X. Li, J. Liu / Discrete Mathematics 309 (2009) 6322–6324
[5] L.B. Kier, L.H. Hall, Molecular Connectivity in Chemistry and Drug Research, Academic Press, New York, 1976. [6] L.B. Kier, L.H. Hall, Molecular Connectivity in Structure-Activity Analysis, Research Studies Press-Wiley, Chichester (UK), 1986. [7] X. Li, I. Gutman, Mathematical Aspects of Randić-Type Molecular Structure Descriptors, Mathematical Chemistry Monographs No.1, Kragujevac, 2006, pp. VI + 330. [8] X. Li, B. Liu, J. Liu, Complete solution to a conjecture on the Randić index, European J. Operational Research, doi:10.1016/j.ejor.2008.12.010 (in press). [9] X. Li, Y. Shi, A survey on the Randić index, MATCH Commun. Math. Comput. Chem. 59 (1) (2008) 127–156. [10] X. Li, Y. Shi, L. Wang, An updated survey on the Randić index, Recent Results in the Theory of Randić Index, Mathematical Chemistry Monograph No.6, Kragujevac, 2008, pp. 9–47. [11] H. Liu, M. Lu, F. Tian, On the Randić index, J. Math. Chem. 38 (3) (2005) 345–354. [12] M. Randić, On characterization of molecular branching, J. Amer. Chem. Soc. 97 (1975) 6609–6615.
Contents lists available at ScienceDirect
Discrete Mathematics journal homepage: www.elsevier.com/locate/disc
Note
Complete solution to a conjecture on the Randić index of triangle-free graphsI Xueliang Li ∗ , Jianxi Liu Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin 300071, PR China
article
info
Article history: Received 4 January 2009 Received in revised form 31 May 2009 Accepted 12 June 2009 Available online 26 June 2009 Keywords: Randić index Conjecture Triangle-free graph
abstract 1
−2 , where d(u) is The Randić index R(G) of a graph G is defined by R(G) = uv (d(u)d(v)) the degree of a vertex u in G and the summation extends over all edges uv of G. A conjecture about the Randić index says that for any √ triangle-free graph G of order n with minimum degree δ ≥ k ≥ 1, one has R(G) ≥ k(n − k), where the equality holds if and only if G = Kk,n−k . In this short note we give a confirmative proof for the conjecture. © 2009 Elsevier B.V. All rights reserved.
P
1. Introduction The Randić index of a graph G was introduced by the chemist Milan Randić in 1975, which is defined by R(G) =
X 1 (d(u)d(v))− 2 , uv
where d(u) is the degree of a vertex u in G and the summation extends over all edges uv of G. This topological index is suitable for measuring the extent of branching of the carbon-atom skeleton of saturated hydrocarbons. Randić himself demonstrated [12] that his index is well correlated with a variety of physico-chemical properties of alkanes. Nowadays the index becomes one of the most popular molecular descriptors √ to which three books are devoted [5–7]. Bollobás and Erdős [1] gave the sharp lower bound R(G) ≥ n − 1 for G being a graph of order n with minimum degree at least 1. In [3] Fajtlowitcz mentioned that Bollobás and Erdős asked for the minimum value on the Randić index among the graphs with given minimum degree δ . Delorme, Favaron and Rautenbach [2] answered this question for δ = 2 and proposed a conjecture concerning the minimum value of the index R for all graphs of order n with minimum degree δ . They also gave a best possible lower bound for the index R of a triangle-free graph with given lower bound of the minimum degree δ : Let G be a triangle-free graph of order n with minimum degree δ ≥ k ≥ 1. Then R(G) ≥
p
k(n − k),
where the equality holds if and only if G = Kk,n−k . Liu, Lu and Tian [11] pointed out a mistake in the proof of this result, and gave a new proof only for the case when k = 2. It is proposed as a conjecture by one of the present authors in [7,9,10] as follows:
I Supported by NSFC No.10831001, PCSIRT and the ‘‘973’’ program.
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (X. Li).
0012-365X/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2009.06.003
X. Li, J. Liu / Discrete Mathematics 309 (2009) 6322–6324
6323
Conjecture 1.1. Let G be a triangle-free graph of order n with minimum degree δ ≥ k ≥ 1. Then R(G) ≥
p
k(n − k),
where the equality holds if and only if G = Kk,n−k . In this paper, we want to prove that Conjecture 1.1 is true, completely solving the conjecture. 2. Main result The following result was obtained by Favaron et al. in [4], and is need in what follows. Lemma 2.1 ([4]). For any triangle-free graph G with m edges, we have R(G) ≥
√
m.
Now we come to the proof of Conjecture 1.1. Theorem 2.2. For any triangle-free graph G of order n and minimum degree δ ≥ k ≥ 1, we have R(G) ≥
p
k(n − k).
Equality holds if and only if G = Kk,n−k . Proof. Let m denote the number of edges of G and let xi,j denote the number of edges uv of G with {dG (u), dG (v)} = {i, j}. If m > k(n − k), then the result follows from Lemma 2.1 and equality is not possible. Hence we may assume m ≤ k(n − k). Since G is triangle-free, adjacent vertices have disjoint neighborhoods which imply that ∆ ≤ n − δ . From Eq. (2.1.13) of [7] we know that the Randić index can be rewritten as R =
≥
=
n 2 n 2 n
−
−
2 δ≤i≤j≤∆ 1
X
2 δ≤i≤j≤n−δ 1
1
j
i
1
1
2
√ −√ n−δ δ 2 X
xi,j
1
xi,j √ −√ n−δ δ δ≤i≤j≤n−δ 2 1 n m 1 = − √ −√ 2 2 n−δ δ 2 k(n − k) 1 1 n ≥ − √ −√ 2 2 n−δ δ 2 p n k(n − k) 1 1 ≥ − = k(n − k). √ −√ 2 2 n−k k 2
−
2 X 1 1 − xi,j √ √
1
2
In the above inequality chain equality holds throughout if and only if m = k(n − k) = xk,n−k , which immediately implies that G = Kk,n−k . Remark. Li, Liu and Liu [8] completely solved the problem for the minimum value on the Randić index for the graphs with given minimum degree δ . Theorem 2.2 in the present paper completely solves the conjecture for graphs without triangles. Up until now, the conjectures proposed in [2] have been completely solved. Acknowledgements The authors would like to express thanks to two anonymous referees for detailed comments and suggestions, which helped to shorten the paper dramatically. References [1] [2] [3] [4]
B. Bollobás, P. Erdős, Graphs of extremal weights, Ars Combin. 50 (1998) 225–233. C. Delorme, O. Favaron, D. Rautenbach, On the Randić index, Discrete Math. 257 (1) (2002) 29–38. S. Fajtlowicz, Written on the wall, conjectures derived on the basis of the program Galatea Gabriella Graffiti, University of Houston, 1998. O. Favaron, M. Mahéo, J.F. Saclé, Some eigenvalue properties in graphs (Conjecture of Graffiti-II), Discrete Math. 111 (1–3) (1993) 197–220.
6324
X. Li, J. Liu / Discrete Mathematics 309 (2009) 6322–6324
[5] L.B. Kier, L.H. Hall, Molecular Connectivity in Chemistry and Drug Research, Academic Press, New York, 1976. [6] L.B. Kier, L.H. Hall, Molecular Connectivity in Structure-Activity Analysis, Research Studies Press-Wiley, Chichester (UK), 1986. [7] X. Li, I. Gutman, Mathematical Aspects of Randić-Type Molecular Structure Descriptors, Mathematical Chemistry Monographs No.1, Kragujevac, 2006, pp. VI + 330. [8] X. Li, B. Liu, J. Liu, Complete solution to a conjecture on the Randić index, European J. Operational Research, doi:10.1016/j.ejor.2008.12.010 (in press). [9] X. Li, Y. Shi, A survey on the Randić index, MATCH Commun. Math. Comput. Chem. 59 (1) (2008) 127–156. [10] X. Li, Y. Shi, L. Wang, An updated survey on the Randić index, Recent Results in the Theory of Randić Index, Mathematical Chemistry Monograph No.6, Kragujevac, 2008, pp. 9–47. [11] H. Liu, M. Lu, F. Tian, On the Randić index, J. Math. Chem. 38 (3) (2005) 345–354. [12] M. Randić, On characterization of molecular branching, J. Amer. Chem. Soc. 97 (1975) 6609–6615.