Related papers: Distances on the tropical line determined by two p…
A linear forest is a collection of vertex-disjoint paths. The Linear Arboricity Conjecture states that every graph of maximum degree $\Delta$ can be decomposed into at most $\lceil(\Delta+1)/2\rceil$ linear forests. We prove that $\Delta/2…
We develop a tropical analogue of the classical double description method allowing one to compute an internal representation (in terms of vertices) of a polyhedron defined externally (by inequalities). The heart of the tropical algorithm is…
A net is a special configuration of lines and points in the projective plane. There are certain restrictions on the number of its lines and points. We proved that there cannot be any (4,4) nets in $\mathbb{C}P^2$. In order to show this, we…
Tropical geometry gives a bound on the ranks of divisors on curves in terms of the combinatorics of the dual graph of a degeneration. We show that for a family of examples, curves realizing this bound might only exist over certain…
Maxmin trees are labeled trees with the property that each vertex is either a local maximum or a local minimum. Such trees were originally introduced by Postnikov, who gave a formula to count them and different combinatorial interpretations…
We study the behavior of phylogenetic tree shapes in the tropical geometric interpretation of tree space. Tree shapes are formally referred to as tree topologies; a tree topology can also be thought of as a tree combinatorial type, which is…
We compute a number of distance-dependent universal scaling functions characterizing the distance statistics of large maps of genus one. In particular, we obtain explicitly the probability distribution for the length of the shortest…
{\it .}We completely characterize pairs of lattice points $P_1\neq P_2$ in the plane with the property that there are infinitely many lattice points $Q$ whose distance from both $P_1$ and $P_2$ is integral. In particular we show that it…
The problem of how to estimate diffusion on a graph effectively is of importance both theoretically and practically. In this paper, we make use of two widely studied indices, geodesic distance and mean first-passage time ($MFPT$) for random…
This paper investigates the geometric properties of a special case of the two-sided system given by $2 \times 2$ tropical commuting constraints. Given a finite matrix $A \in \mathbb{R}^{2\times 2}$, the paper studies the extremals of the…
We present tools and definitions to study abstract tropical manifolds in dimension 2, which we call simply tropical surfaces. This includes explicit descriptions of intersection numbers of 1-cycles, normal bundles to some curves and…
We develop the rudiments of a finite-dimensional representation theory of groups over idempotent semifields by considering linear actions on tropical linear spaces. This can be considered a tropical representation theory, a characteristic…
The classic Maxwell formula calculates the length of a planar locally minimal binary tree in terms of coordinates of its boundary vertices and directions of incoming edges. However, if an extreme tree with a given topology and a boundary…
There are several tools available to infer phylogenetic trees, which depict the evolutionary relationships among biological entities such as viral and bacterial strains in infectious outbreaks, or cancerous cells in tumor progression trees.…
There are several interrelated notions of discrete curvature on graphs. Many approaches utilize the optimal transportation metric on its probability simplex or the distance matrix of the graph. In this survey article, we compute formulas…
As well known the rotation distance D(S,T) between two binary trees S, T of n vertices is the minimum number of rotations of pairs of vertices to transform S into T. We introduce the new operation of chain rotation on a tree, involving two…
In this paper we study general tropical linear spaces locally: For any basis B of the matroid underlying a tropical linear space L, we define the local tropical linear space L_B to be the subcomplex of L consisting of all vectors v that…
Understanding the evolution of a set of genes or species is a fundamental problem in evolutionary biology. The problem we study here takes as input a set of trees describing {possibly discordant} evolutionary scenarios for a given set of…
Let $f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}$, let $(a,b,c,d)$ stand for $a,b,c,d\in\mathbb Z_{\geq 0}$ such that $ad-bc=1$. Define \begin{equation} \label{eq_main} F(s) = \sum_{(a,b,c,d)} f(a,b,c,d)^s.…
In analogy to other concepts of a similar nature, we define the inducibility of a rooted binary tree. Given a fixed rooted binary tree $B$ with $k$ leaves, we let $\gamma(B,T)$ be the proportion of all subsets of $k$ leaves in $T$ that…