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

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(S_n^k)) ≤ W(μ(T_n^k)) ≤ W(μ(P_n^k)), where S_n, T_n and P_n 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(G^k).

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

2010 Mathematics Subject Classification: 05C12.

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