Discussiones Mathematicae Graph Theory 30(3) (2010) 489-498
doi: 10.7151/dmgt.1509

[BIBTex] [PDF] [PS]

THE WIENER NUMBER OF POWERS OF THE MYCIELSKIAN

Rangaswami Balakrishnan  and  S. Francis Raj

Srinivasa Ramanujan Centre
SASTRA University
Kumbakonam-612 001, India
e-mail: mathbala@satyam.net.in
e-mail: francisraj_s@yahoo.com

Abstract

The Wiener number of a graph G is defined as [1/2] ∑u,v ∈ V(G)d(u, v), d the distance function on G. The Wiener number has important applications in chemistry. We determine a formula for the Wiener number of an important graph family, namely, the Mycielskians μ(G) of graphs G. Using this, we show that for k ≥ 1, W(μ(Snk)) ≤ W(μ(Tnk)) ≤ W(μ(Pnk)), where Sn, Tn and Pn denote a star, a general tree and a path on n vertices respectively. We also obtain Nordhaus-Gaddum type inequality for the Wiener number of μ(Gk).

Keywords: Wiener number, Mycielskian, powers of a graph.

2010 Mathematics Subject Classification: 05C12.

References

[1] X. An and B. Wu, The Wiener index of the kth power of a graph, Appl. Math. Lett. 21 (2007) 436-440, doi: 10.1016/j.aml.2007.03.025.
[2] R. Balakrishanan and S.F. Raj, The Wiener number of Kneser graphs, Discuss. Math. Graph Theory 28 (2008) 219-228, doi: 10.7151/dmgt.1402.
[3] R. Balakrishanan, N. Sridharan and K.V. Iyer, Wiener index of graphs with more than one cut vertex, Appl. Math. Lett. 21 (2008) 922-927, doi: 10.1016/j.aml.2007.10.003.
[4] R. Balakrishanan, N. Sridharan and K.V. Iyer, A sharp lower bound for the Wiener Index of a graph, to appear in Ars Combinatoria.
[5] R. Balakrishanan, K. Viswanathan and K.T. Raghavendra, Wiener Index of Two Special Trees, MATCH Commun. Math. Comput. Chem. 57 (2007) 385-392.
[6] G.J. Chang, L. Huang and X. Zhu, Circular Chromatic Number of Mycielski's graphs, Discrete Math. 205 (1999) 23-37, doi: 10.1016/S0012-365X(99)00033-3.
[7] A.A. Dobrynin, I. Gutman, S. Klavžar and P. Zigert, Wiener Index of Hexagonal Systems, Acta Appl. Math. 72 (2002) 247-294, doi: 10.1023/A:1016290123303.
[8] H. Hajibolhassan and X. Zhu, The Circular Chromatic Number and Mycielski construction, J. Graph Theory 44 (2003) 106-115, doi: 10.1002/jgt.10128.
[9] D. Liu, Circular Chromatic Number for iterated Mycielski graphs, Discrete Math. 285 (2004) 335-340, doi: 10.1016/j.disc.2004.01.020.
[10] Liu Hongmei, Circular Chromatic Number and Mycielski graphs, Acta Mathematica Scientia 26B (2006) 314-320.
[11] J. Mycielski, Sur le colouriage des graphes, Colloq. Math. 3 (1955) 161-162.
[12] E.A. Nordhaus and J.W. Gaddum, On complementary graphs, Amer. Math. Monthly 63 (1956) 175-177, doi: 10.2307/2306658.
[13] H. Wiener, Structural Determination of Paraffin Boiling Points, J. Amer. Chem. Soc. 69 (1947) 17-20, doi: 10.1021/ja01193a005.
[14] L. Xu and X. Guo, Catacondensed Hexagonal Systems with Large Wiener Numbers, MATCH Commun. Math. Comput. Chem. 55 (2006) 137-158.
[15] L. Zhang and B. Wu, The Nordhaus-Gaddum-type inequalities for some chemical indices, MATCH Commun. Math. Comput. Chem. 54 (2005) 189-194.

Received 14 November 2008
Revised 8 October 2009
Accepted 20 October 2009