Related papers: A hyponormal weighted shift on a directed tree who…
We study rational functions $f$ of degree $d+1$ such that $f$ is univalent in the exterior unit disc, and the image of the unit circle under $f$ has the maximal number of cusps ($d+1$) and double points $(d-2)$. We introduce a bi-angled…
We consider planar rooted random trees whose distribution is even for fixed height $h$ and size $N$ and whose height dependence is of exponential form $e^{-\mu h}$. Defining the total weight for such trees of fixed size to be $Z^{(\mu)}_N$,…
We systematically develop the multivariable counterpart of the theory of weighted shifts on rooted directed trees. Capitalizing on the theory of product of directed graphs, we introduce and study the notion of multishifts on directed…
Dynamic trees are mixtures of tree structured belief networks. They solve some of the problems of fixed tree networks at the cost of making exact inference intractable. For this reason approximate methods such as sampling or mean field…
Hadwiger's conjecture asserts that any graph contains a clique minor with order no less than the chromatic number of the graph. We prove that this well-known conjecture is true for all graphs if and only if it is true for squares of split…
We consider problems in which we are given a rooted tree as input, and must find a subtree with the same root, optimizing some objective function of the nodes in the subtree. When this function is the sum of constant node weights, the…
This paper studies the "explanation problem" for tree- and linearly-ordered array data, a problem motivated by database applications and recently solved for the one-dimensional tree-ordered case. In this paper, one is given a matrix A whose…
We initiate the study of the forward and backward shifts on the Lipschitz space of a tree, $\mathcal L$, and on the little Lipshitz space of a tree, ${\mathcal L}_0$. We determine that the forward shift is bounded both on $\mathcal L$ and…
This paper is devoted to the study of propagation phenomena for $2$--hyponormal, quadratically hyponormal, and cubically hyponormal operator-valued weighted shifts. \ First, we show that every {\it quadratically} hyponormal matrix-valued…
We survey the current state of progress on the Graceful Tree Conjecture, and then we present several new results toward the conjecture, driven by three new ideas: (1) It has been proven that generalized banana trees are graceful by…
We prove that every oriented tree on $n$ vertices with bounded maximum degree appears as a spanning subdigraph of every directed graph on $n$ vertices with minimum semidegree at least $n/2+o(n)$. This can be seen as a directed graph…
In this paper we provide some exact formulas for projective dimension and the regularity of powers of edge ideals of vertex-weighted rooted forests. These formulas are functions of the weight of the vertices and the number of edges. We also…
Let $\mathcal{B}$ be the set of rooted trees containing an infinite binary subtree starting at the root. This set satisfies the metaproperty that a tree belongs to it if and only if its root has children $u$ and $v$ such that the subtrees…
In this paper we investigate hereditarily normal topological groups and their subspaces. We prove that every compact subspace of a hereditarily normal topological group is metrizable. To prove this statement we first show that a…
We study jointly quasinormal and spherically quasinormal pairs of commuting operators on Hilbert space, as well as their powers. We first prove that, up to a constant multiple, the only jointly quasinormal $2$-variable weighted shift is the…
This paper, dating from May 1991, contains preliminary (and unpublishable) notes on investigations about iteration trees. They will be of interest only to the specialist. In the first two sections I define notions of support and embeddings…
We consider a Gibbs distribution over all spanning trees of an undirected, edge weighted finite graph, where, up to normalization, the probability of each tree is given by the product of its edge weights. Defining the weighted degree of a…
We classify weakly connected spanning closed (WCSC) subgraphs of $\overrightarrow{C_n^2}$, the square of a directed $n$-vertex cycle. Then we show that every spanning tree of $\overrightarrow{C_n^2}$ is contained in a unique nontrivial WCSC…
We consider three different schemes for signal routing on a tree. The vertices of the tree represent transceivers that can transmit and receive signals, and are equipped with i.i.d. weights representing the strength of the transceivers. The…
Rotation distance between trees measures the number of simple operations it takes to transform one tree into another. There are no known polynomial-time algorithms for computing rotation distance. In the case of ordered rooted trees, we…