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In 1997 Klav\v{z}ar and Gutman suggested a generalization of the Wiener indexto vertex-weighted graphs. We minimize the Wiener index over the set of treeswith the given vertex weights' and degrees' sequences and show an optimal treeto be the, so-called, Huffman tree built in a bottom-up manner by sequentiallyconnecting vertices of the least weights.

Minimizing Wiener Index for Vertex-Weighted Trees with Given Weight and Degree Sequences