Wiener Dimension: Fundamental Properties and (5,0)-Nanotubical Fullerenes

The Wiener dimension of a connected graph is introduced as the number of different 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 nonvertex-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. Corresponding author

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