Related papers: Active spanning trees and Schramm-Loewner evolutio…
We show that the diameter of a uniformly drawn spanning tree of a connected graph on $n$ vertices which satisfies certain high-dimensionality conditions typically grows like $\Theta(\sqrt{n})$. In particular this result applies to…
We give bijections between bipolar-oriented (acyclic with unique source and sink) planar maps and certain random walks, which show that the uniformly random bipolar-oriented planar map, decorated by the "peano curve" surrounding the tree of…
We consider the problem of uniformly generating a spanning tree, of a connected undirected graph. This process is useful to compute statistics, namely for phylogenetic trees. We describe a Markov chain for producing these trees. For cycle…
A $(1+\varepsilon)$-stretch tree cover of an edge-weighted $n$-vertex graph $G$ is a collection of trees, where every pair of vertices has a $(1+\varepsilon)$-stretch path in one of the trees. The celebrated Dumbbell Theorem by Arya et. al.…
We study the crossing-minimization problem in a layered graph drawing of planar-embedded rooted trees whose leaves have a given total order on the first layer, which adheres to the embedding of each individual tree. The task is then to…
The quadratic minimum spanning tree problem and its variations such as the quadratic bottleneck spanning tree problem, the minimum spanning tree problem with conflict pair constraints, and the bottleneck spanning tree problem with conflict…
The Tutte polynomial of a graph is a 2-variable polynomial which is quite important in both combinatorics and statistical physics. It contains various numerical invariants and polynomial invariants, such as the number of spanning trees, the…
It is well-known that the number of spanning trees, denoted by $\tau(G)$, in a connected multi-graph $G$ can be calculated by the Matrix-Tree theorem and Tutte's deletion-contraction theorem. In this short note, we find an alternate method…
For a two-dimensional lattice $\Lambda$ with $n$ vertices, the number of spanning trees $N_{ST}(\Lambda)$ grows asymptotically as $\exp(n z_\Lambda)$ in the thermodynamic limit. We present exact integral expression and numerical value for…
Tree-width and path-width are well-known graph parameters. Many NP-hard graph problems allow polynomial-time solutions, when restricted to graphs of bounded tree-width or bounded path-width. In this work, we study the behavior of tree-width…
The Shcramm-Loewner evolution (SLE) is a correlated exploration process, in which for the chordal set up, the tip of the trace evolves in a self-avoiding manner towards the infinity. The resulting curves are named SLE$_{\kappa}$,…
Consider the nearest neighbor graph for the integer lattice Z^d in d dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees. As the piece…
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…
Let $\ell$ and $p$ be two distinct primes. We study the $p$-adic valuation of the number of spanning trees in an abelian $\ell$-tower of connected multigraphs. This is analogous to the classical theorem of Washington--Sinnott on the growth…
Expansion property of a graph refers to its strong connectivity as well as sparseness. It has been reported that deep neural networks can be pruned to a high degree of sparsity while maintaining their performance. Such pruning is essential…
A spanning generalized caterpillar is a spanning tree in which all vertices of degree more than two are on a path. In this note, we find a relation between the existence of spanning generalized caterpillar and the independence and…
For a simple drawing $D$ of the complete graph $K_n$, two (plane) subdrawings are compatible if their union is plane. Let $\mathcal{T}_D$ be the set of all plane spanning trees on $D$ and $\mathcal{F}(\mathcal{T}_D)$ be the compatibility…
Schramm-Loewner Evolutions ($\SLE$) are random curves in planar simply connected domains; the massless (Euclidean) free field in such a domain is a random distribution. Both have conformal invariance properties in law. In the present…
We consider the polyhedral properties of two spanning tree problems with additional constraints. In the first problem, it is required to find a tree with a minimum sum of edge weights among all spanning trees with the number of leaves less…
Connectivity query processing is a fundamental problem in graph processing. Given an undirected graph and two query vertices, the problem aims to identify whether they are connected via a path. Given frequent edge updates in real graph…