Computing quadratic entropy in evolutionary trees

We note here that quadratic entropy, a measure of biological diversity introduced by C.R. Rao, is a variant of the weighted Wiener index, a graph invariant intensively studied in mathematical chemistry. This fact allows us to deduce some efficient algorithms for computing the quadratic entropy in the case of given tip weights, which may be useful for community biodiversity measures. Furthermore, on ultrametric phylogenetic trees, the maximum of quadratic entropy is a measure of pairwise evolutionary distinctness in conservation biology, introduced by S. Pavoine. We present an algorithm that maximizes this quantity in linear time, offering a significant improvement over the currently used quadratic programming approaches.