Related papers: Increasing Trees and the Degree-Chromatic Polynomi…
When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…
Tanglegrams are a special class of graphs appearing in applications concerning cospeciation and coevolution in biology and computer science. They are formed by identifying the leaves of two rooted binary trees. We give an explicit formula…
In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general…
In this paper we prove that the dominator chromatic number of every oriented tree is invariant under reversal of orientation. In addition to this marquee result, we also prove the exact dominator chromatic number for arborescences and…
Using the special value at $u=1$ of Artin-Ihara $L$-functions, we associate to every $\mathbb{Z}$-cover of a finite connected graph a polynomial which we call the \emph{Ihara polynomial}. We show that the number of spanning trees for the…
We study the fundamental question of how likely it is that two randomly chosen trees are isomorphic to each other for different models of random trees. We show that the probability decays exponentially for rooted labeled trees as well as…
The study of patterns in permutations associated with forests of binary shrubs was initiated by D. Bevan et al.. In this paper, we study five different types of rise statistics that can be associated with such permutations and find the…
Several hook summation formulae for binary trees have appeared recently in the literature. In this paper we present an analogous formula for unordered increasing trees of size r, which involves r parameters. The right-hand side can be…
In this article, associated to a (bordered) Legendrian graph, we study and show the equivalence between two Legendrian isotopy invariants: augmentation number via point-counting over a finite field, for the augmentation variety of the…
Ma-Ma-Yeh made a beautiful observation that a transformation of the grammar of Dumont instantly leads to the $\gamma$-positivity of the Eulerian polynomials. We notice that the transformed grammar bears a striking resemblance to the grammar…
We give a short and direct proof of a remarkable identity that arises in the enumeration of labeled trees with respect to their indegree sequence, where all edges are oriented from the vertex with lower label towards the vertex with higher…
We establish a connection between root multiplicities for Borcherds-Kac-Moody algebras and graph coloring. We show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed…
We study the virial expansion of mixtures of countably many different types of particles. The main tool is the Lagrange-Good inversion formula, which has other applications such as counting coloured trees or studying probability generating…
The in-order traversal provides a natural correspondence between binary trees with a decreasing vertex labeling and endofunctions on a finite set. By suitably restricting the vertex labeling we arrive at a class of trees that we call…
In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erd\H{o}s-R\'enyi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every…
We generalize the concept of ascending and descending runs from permutations to rooted labelled trees and mappings, i.e., functions from the set $\{1, \dots, n\}$ into itself. A combinatorial decomposition of the corresponding functional…
Weighted recursive trees are built by adding successively vertices with predetermined weights to a tree: each new vertex is attached to a parent chosen at random with probability proportional to its weight. In the case where the total…
Counting non-isomorphic tree-like multigraphs that include self-loops and multiple edges is an important problem in combinatorial enumeration, with applications in chemical graph theory, polymer science, and network modeling. Traditional…
For a labeled tree on the vertex set $\set{1,2,\ldots,n}$, the local direction of each edge $(i\,j)$ is from $i$ to $j$ if $i<j$. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges…
An increasing 1,2-tree is a labeled graph formed by starting with a vertex and then repeatedly attaching a leaf to a vertex or a triangle to an edge, the labeling of the vertices corresponding to the order in which the vertices are added.…