Entropy measures of distance matrix

In this paper, two distance-based entropy measures are studied that were introduced by Bonchev and Trinajstić [J. Chem. Phys. 38 (1977) 4517–4533] for interpreting the molecular branching of molecular graphs. One of these entropy measures is based on the distribution of distances in the distance matrix and the other one is based on the distribution of distances in the upper triangular submatrix. The mentioned measures are calculated for paths, stars, complete graphs, cycles, and complete bipartite graphs. The minimal trees with respect to the entropy measures under consideration with fixed diameter are also determined.