Related papers: Postnikov-Shapiro Algebras, Graphical Matroids and…
For a connected labelled graph $G$, a {\em spanning tree} $T$ is a connected and an acyclic subgraph that spans all vertices of $G$. In this paper, we consider a classical combinatorial problem which is to list all spanning trees of $G$. A…
One of the main virtues of trees is to represent formal solutions of various functional equations which can be cast in the form of fixed point problems. Basic examples include differential equations and functional (Lagrange) inversion in…
Generalized Pythagoras trees were developed for visualizing hierarchical data, producing organic, fractal-like representations. However, the drawback of the original layout algorithm is visual overlap of tree branches. To avoid such…
We construct and study new generalisations to rooted trees and forests of some properties of shuffles of words. First, we build a coproduct on rooted trees which, together with their shuffle, endow them with bialgebra structure. We then…
This paper provides the generating series for the embedding of tree-like graphs of arbitrary number of vertices, accourding to their genus. It applies and extends the techniques of Chan, where it was used to give an alternate proof of the…
In this paper we study several problems concerning the number of homomorphisms of trees. We give an algorithm for the number of homomorphisms from a tree to any graph by the Transfer-matrix method. By using this algorithm and some…
Measuring the complexity of tree structures can be beneficial in areas that use tree data structures for storage, communication, and processing purposes. This complexity can then be used to compress tree data structures to their…
We present new short proofs of known spanning tree enumeration formulae for threshold and Ferrers graphs by showing that the Laplacian matrices of such graphs admit triangular rank-one perturbations. We then characterize the set of graphs…
Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets.…
A traversal of a connected graph is a linear ordering of its vertices all of whose initial segments induce connected subgraphs. Traversals, and their refinements such as breadth-first and depth-first traversals, are computed by various…
If $G$ is a strongly connected finite directed graph, the set $\mathcal{T}G$ of rooted directed spanning trees of $G$ is naturally equipped with a structure of directed graph: there is a directed edge from any spanning tree to any other…
Let ${\rm Z}(G)$ and ${\rm gp}(G)$ be the zero forcing number and the general position number of a graph $G$, respectively. Known results imply that ${\rm gp}(T)\ge {\rm Z}(T) + 1$ holds for every nontrivial tree $T$. It is proved that the…
Using local detailed balance we rewrite the Kirchhoff formula for stationary distribution of Markov jump processes in terms of a physically interpretable tree-ensemble. We use that arborification of path-space integration to derive a…
The matrices of spanning rooted forests are studied as a tool for analysing the structure of digraphs and measuring their characteristics. The problems of revealing the basis bicomponents, measuring vertex proximity, and ranking from…
The Laplacian matrix of a graph $G$ is $L(G)=D(G)-A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the diagonal matrix of vertex degrees. According to the Matrix-Tree Theorem, the number of spanning trees in $G$ is equal to any…
We define an algebraic setup of homology for hypergraphs, which defaults to simplicial homology in the case of graphs, and study its basic properties. As part of our study we define algebraic spanning trees of hypergraphs, along with…
This paper presents a new approach for trees-based regression, such as simple regression tree, random forest and gradient boosting, in settings involving correlated data. We show the problems that arise when implementing standard…
We prove that a polynomial fraction of the set of $k$-component forests in the $m \times n$ grid graph have equal numbers of vertices in each component, for any constant $k$. This resolves a conjecture of Charikar, Liu, Liu, and Vuong, and…
We propose a novel definition of hypergraphical matroids, defined for arbitrary hypergraphs, simultaneously generalizing previous definitions for regular hypergraphs (Main, 1978), and for the hypergraphs of circuits of a matroid…
We give an algorithm for finding the arboricity of a weighted, undirected graph, defined as the minimum number of spanning forests that cover all edges of the graph, in $\sqrt{n} m^{1+o(1)}$ time. This improves on the previous best bound of…