相关论文: Boundary Partitions in Trees and Dimers

A closed-form formula is derived for the number of occurrences of matches of a multiset of patterns among all ordered (plane-planted) trees with a given number of edges. A pattern looks like a tree, with internal nodes and leaves, but also…

With the algebraic trees, L\"ohr and Winter (2021) introduced a generalization of the notion of graph-theoretic trees to account for potentially uncountable structures. The tree structure is given by the map which assigns to each triple of…

A class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and…

The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the…

We investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of…

We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding…

This work addresses the intrinsic relationship between trees and networks (i.e. graphs). A complete (invertible) mapping is presented which allows trees to be mapped into weighted graphs and then backmapped into the original tree without…

We discuss two sampling schemes for selecting random subnets from a network: Random sampling and connectivity dependent sampling, and investigate how the degree distribution of a node in the network is affected by the two types of sampling.…

A binary tanglegram is a pair <S,T> of binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example in phylogenetics or software engineering, it is required…

A tanglegram consists of two rooted binary trees and a perfect matching between their leaves, and a planar tanglegram is one that admits a layout with no crossings. We show that the problem of generating planar tanglegrams uniformly at…

We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for polytopes defined by integer constraints in terms of the height of the integers and certain subdeterminants of the constraint…

Phylogenetic networks generalise phylogenetic trees and allow for the accurate representation of the evolutionary history of a set of present-day species whose past includes reticulate events such as hybridisation and lateral gene transfer.…

Probabilistic circuits (PCs) have emerged as a powerful framework to compactly represent probability distributions for efficient and exact probabilistic inference. It has been shown that PCs with a general directed acyclic graph (DAG)…

In this paper we investigate the geometry of a discrete Bayesian network whose graph is a tree all of whose variables are binary and the only observed variables are those labeling its leaves. We provide the full geometric description of…

We study the query complexity of determining if a graph is connected with global queries. The first model we look at is matrix-vector multiplication queries to the adjacency matrix. Here, for an $n$-vertex graph with adjacency matrix $A$,…

Divided symmetrization of a function $f(x_1,\dots,x_n)$ is symmetrization of the ratio $$DS_G(f)=\frac{f(x_1,\dots,x_n)}{\prod (x_i-x_j)},$$ where the product is taken over the set of edges of some graph $G$. We concentrate on the case when…

An independent set in a graph is a set of pairwise non-adjacent vertices. Let $\alpha(G)$ denote the cardinality of a maximum independent set in the graph $G = (V, E)$. Gutman and Harary defined the independence polynomial of $G$ \[ I(G;x)…

For an integer $k$, a $k$-tree is a tree with maximum degree at most $k$. More generally, if $f$ is an integer-valued function on vertices, an $f$-tree is a tree in which each vertex $v$ has degree at most $f(v)$. Let $c(G)$ denote the…

Correlation functions in quantum field theory are calculated using Feynman amplitudes, which are finite dimensional integrals associated to graphs. The integrand is the exponential of the ratio of the first and second Symanzik polynomials…

Schnyder woods are decompositions of simple triangulations into three edge-disjoint spanning trees crossing each other in a specific way. In this article, we define a generalization of Schnyder woods to $d$-angulations (plane graphs with…