Wiener Dimension: Fundamental Properties and (5,0)-Nanotubical Fullerenes Yaser Alizadeh a , Vesna Andova b1 , Sandi Klavˇ zar c,d,e , Riste ˇ Skrekovski c,e,f,g a Department of Mathematics, Hakim Sabzevari University, Sabzevar, Iran E-mail: y.alizadeh@hsu.ac.ir 2 Faculty of Electrical Engineering and Information Technologies, Ss Cyril and Methodius Univ., Ruger Boskovik, P. O. Box 574, 1000 Skopje, Macedonia E-mail: vesna.andova@gmail.com c Faculty of Mathematics and Physics, University of Ljubljana, Slovenia d Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia e Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia E-mail: sandi.klavzar@fmf.uni-lj.si f Faculty of Information Studies, Novo Mesto, Slovenia g Faculty of Mathematics, Natural Sciences and Information Technologies, University of Primorska, Slovenia E-mail: skrekovski@gmail.com (Received January 24, 2014) Abstract The Wiener dimension of a connected graph is introduced as the number of dif- ferent distances of its vertices. For any integer D and any integer k, a graph of diameter D and of Wiener dimension k is constructed. An infinite family of non- vertex-transitive graphs with Wiener dimension 1 is presented and it is proved that a graph of dimension 1 is 2-connected. It is shown that the (5, 0)-nanotubical fullerene graph on 10k (k ≥ 3) vertices has Wiener dimension k. As a consequence the Wiener index of these fullerenes is obtained. 1 Corresponding author