Abstract
AbstractThe Szeged index is a graph invariant which is a natural generalization of Wienerindex. In this note, we disprove two recent conjectures concerning with the maxi-mum value of Szeged index of graphs, which are due to Khalifeh et al. (Europ. J.Combinatorics (2008), doi:10.1016/j.ejc.2008.09.019) and respectively, to Gutmanet al. (Groat. Chem. Acta 81(2)(2008) 263-266) and prove a conjecture on Szegedindex due to Klavzar et al. ( Appl. Math. Lett. 9(1996), 45-49), which states thatthe complete bipartite graph K n2 , n2 has maximum Szeged index among all con-nected graphs on n vertices. The last conjecture is previously proved by Dobrynin(Croat. Chem. Acta 70(3), 819-825), but our proof turns out to be much simplerand self-contained.Keywords: Szeged index, Wiener index 1 Introduction All graphs considered here are finite, undirected and simple. We refer to [4] forunexplained terminology and notations. Let G =(V(G),E(G)) be a graph. 1 Supported by Canada Research Chair Program Electronic Notes in Discrete Mathematics 34 (2009) 405–4091571-0653/$ – see front matter Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved.www.elsevier.com/locate/endmdoi:10.1016/j.endm.2009.07.067
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