Related papers: Hyperforests on the Complete Hypergraph by Grassma…
In this paper, we explore some interesting applications of the matrix tree theorem. In particular, we present a combinatorial interpretation of a distribution of $(n-1)^{n-1}$, in the context of uprooted spanning trees of the complete graph…
We propose a new method for obtaining complete asymptotic expansions in a systematic manner, which is suitable for counting sequences of various graph families in dense regime. The core idea is to encode the two-dimensional array of…
The recursive and hierarchical structure of full rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is…
Given a collection of n rooted trees with depth h, we give a necessary and sufficient condition for this collection to be the collection of h-depth universal covering neighborhoods at each vertex.
Let $A$ be a commutative $k$-algebra over a field of $k$ and $\Xi$ a linear operator defined on $A$. We define a family of $A$-valued invariants $\Psi$ for finite rooted forests by a recurrent algorithm using the operator $\Xi$ and show…
Graph polynomials encode fundamental combinatorial invariants of graphs. Their computation is investigated using tree and path decomposition frameworks, with formal definitions of treewidth, k-trees, and pathwidth establishing the…
In a previous paper, we showed that a compartmental stochastic process model of SARS-CoV-2 transmission could be fit to time series data and then reinterpreted as a collection of interacting branching processes drawn from a dynamic degree…
We study the enumeration problem for different kind of tree parking functions introduced recently, called tree parking functions, tree parking distributions, prime tree parking functions, and prime tree parking distributions, for rooted…
Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely…
We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call…
We extend Edmonds' Branching Theorem to locally finite infinite digraphs. As examples of Oxley or Aharoni and Thomassen show, this cannot be done using ordinary arborescences, whose underlying graphs are trees. Instead we introduce the…
We study a model of growing planar tree graphs where in each time step we separate the tree into two components by splitting a vertex and then connect the two pieces by inserting a new link between the daughter vertices. This model…
We study random unrooted plane trees with $n$ vertices sampled according to the weights corresponding to the vertex-degrees. Our main result shows that if the generating series of the weights has positive radius of convergence, then this…
In this article, we extend Moon's classic formula for counting spanning trees in complete graphs containing a fixed spanning forest to complete bipartite graphs. Let $(X,Y)$ be the bipartition of the complete bipartite graph $K_{m,n}$ with…
We revisit the representation theory in type $A$used previously to establish that the dissimilarity vectors of phylogenetic trees are points on the tropical Grassmannian variety. We use a different version of this construction to show that…
Interpretable machine learning has emerged as central in leveraging artificial intelligence within high-stakes domains such as healthcare, where understanding the rationale behind model predictions is as critical as achieving high…
We compute in small temperature expansion the two-loop renormalization constants and the three-loop coefficient of the beta-function, that is the first non-universal term, for the sigma-model with O(N) invariance on the triangular lattice…
Phylogenetic networks provide a more general description of evolutionary relationships than rooted phylogenetic trees. One way to produce a phylogenetic network is to randomly place $k$ arcs between the edges of a rooted binary phylogenetic…
In phylogenetics, a central problem is to infer the evolutionary relationships between a set of species $X$; these relationships are often depicted via a phylogenetic tree -- a tree having its leaves univocally labeled by elements of $X$…
Inside the discipline of graph theory exists an extension known as the hypergraph. This generalization of graphs includes vertices along with hyperedges consisting of collections of two or more vertices. One well-studied application of this…