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Fermat-Weber points with respect to an asymmetric tropical distance function are studied. It turns out that they correspond to the optimal solutions of a transportation problem. The results are applied to obtain a new method for computing…

It is proposed that the co-expression of statistically significant motifs among the sequences of a proteome is a phylogenetic trait. From the co-expression matrix of such motifs in a group of prokaryotic proteomes a suitable definition of a…

If trees are constructed from a pure birth process and one defines the depth of a leaf to be the number of edges to its root, it is known that the variance in the depth of a randomly selected leaf of a randomly selected tree grows linearly…

In this paper we introduce and study three new measures for efficient discriminative comparison of phylogenetic trees. The NNI navigation dissimilarity $d_{nav}$ counts the steps along a "combing" of the Nearest Neighbor Interchange (NNI)…

Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. They are also objects of interest in pure mathematics, such as algebraic geometry and combinatorics, due to their discrete geometry.…

This paper give a simple linear-time algorithm that, given a weighted digraph, finds a spanning tree that simultaneously approximates a shortest-path tree and a minimum spanning tree. The algorithm provides a continuous trade-off: given the…

Phylogenetic trees are important tools in the study of evolutionary relationships between species. Measures such as the index of Sackin, Colless, and Total Cophenetic have been extensively used to quantify tree balance, one key property of…

We give exact formulas for the transmission (i.e. the sum of all distances between vertices) of perfect trees and rooted powers of (connected finite) graphs.

Graphs drawn in the plane are ubiquitous, arising from data sets through a variety of methods ranging from GIS analysis to image classification to shape analysis. A fundamental problem in this type of data is comparison: given a set of such…

It is a classical result that any finite tree with positively weighted edges, and without vertices of degree 2, is uniquely determined by the weighted path distance between each pair of leaves. Moreover, it is possible for a (small) strict…

For a graph $G=(V,E)$, its exact-distance square, $G^{[\sharp 2]}$, is the graph with vertex set $V$ and with an edge between vertices $x$ and $y$ if and only if $x$ and $y$ have distance (exactly) $2$ in $G$. The graph $G$ is an…

Phylogenetic networks are a generalization of phylogenetic trees that allow for the representation of non-treelike evolutionary events, like recombination, hybridization, or lateral gene transfer. In a recent series of papers devoted to the…

Random forests construct each tree with a different, randomised representation of the feature space. Their uniform voting cannot correct errors in regions where trees with incorrect representations probabilistically outnumber correct ones,…

The Minimal Ancestral Deviation (MAD) method is a recently introduced procedure for estimating the root of a phylogenetic tree, based only on the shape and branch lengths of the tree. The method is loosely derived from the midpoint rooting…

We address an open question of Francis and Steel about phylogenetic networks and trees. They give a polynomial time algorithm to decide if a phylogenetic network, N, is tree-based and pose the problem: given a fixed tree T and network N, is…

The log-det distance between two aligned DNA sequences was introduced as a tool for statistically consistent inference of a gene tree under simple non-mixture models of sequence evolution. Here we prove that the log-det distance, coupled…

Phylogenetic networks are generalizations of trees that allow for the modeling of non-tree like evolutionary processes. Split networks give a useful way to construct networks with intuitive distance structures induced from the associated…

Many popular algorithms for searching the space of leaf-labelled trees are based on tree rearrangement operations. Under any such operation, the problem is reduced to searching a graph where vertices are trees and (undirected) edges are…

We bound the mean distance in a connected graph which is not a tree in function of its order $n$ and its girth $g$. On one hand, we show that mean distance is at most $\frac{n+1}{3}-\frac{g(g^2-4)}{12n(n-1)}$ if $g$ is even and at most…

The Gromov-Hausdorff (GH) distance is a natural way to measure distance between two metric spaces. We prove that it is $\mathrm{NP}$-hard to approximate the Gromov-Hausdorff distance better than a factor of $3$ for geodesic metrics on a…