Related papers: Weighted Join Operators on Directed Trees

Let $G$ be a finitely generated group with polynomial growth, and let $\om$ be a weight, i.e. a sub-multiplicative function on $G$ with positive values. We study when the weighted group algebra $\ell^1(G,\om)$ is isomorphic to an operator…

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…

The Weighted Tree Augmentation Problem (WTAP) is a fundamental well-studied problem in the field of network design. Given an undirected tree $G=(V,E)$, an additional set of edges $L \subseteq V\times V$ disjoint from $E$ called…

We build on recent work of Yeats, Courtiel, and others involving connected chord diagrams. We first derive from a Hopf-algebraic foundation a class of tree-like functional equations and prove that they are solved by weighted generating…

Given a directed graph $G$ and a list $(s_1,t_1),\dots,(s_d,t_d)$ of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of $G$ that contains a directed $s_i\to t_i$ path for every $1\le i \le k$. The…

For a given second-order linear elliptic operator $L$ which admits a positive minimal Green function, and a given positive weight function $W$, we introduce a family of weighted Lebesgue spaces $L^p(\phi_p)$ with their dual spaces, where…

We initiate the algorithmic study of the following "structured augmentation" question: is it possible to increase the connectivity of a given graph G by superposing it with another given graph H? More precisely, graph F is the superposition…

A new method of verifying the subnormality of unbounded Hilbert space operators based on an approximation technique is proposed. Diverse sufficient conditions for subnormality of unbounded weighted shifts on directed trees are established.…

In this paper, we propose a new homological method to study weighted directed networks. Our model of such networks is a directed graph $Q$ equipped with a weight function $w$ on the set $Q_{1}$ of arrows in $Q$. We require that the range…

Let $1 < p < \infty$ and suppose that we are given a function $f$ defined on the leaves of a weighted tree. We would like to extend $f$ to a function $F$ defined on the entire tree, so as to minimize the weighted $W^{1,p}$-Sobolev norm of…

Given an edge weighted graph and a forest $F$, the $\textit{2-edge connectivity augmentation problem}$ is to pick a minimum weighted set of edges, $E'$, such that every connected component of $E'\cup F$ is 2-edge connected. Williamson et…

Let $\mathcal{T}_{+}(E)$ be the tensor algebra of a $W^{*}$-correspondence $E$ over a $W^{*}$-algebra $M$. In earlier work, we showed that the completely contractive representations of $\mathcal{T}_{+}(E)$, whose restrictions to $M$ are…

A resistance network is a weighted graph $(G,c)$ with intrinsic (resistance) metric $R$. We embed the resistance network into the Hilbert space ${\mathcal H}_{\mathcal E}$ of functions of finite energy. We use the resistance metric to study…

In this paper we characterise absolutely norm attaining quasi*paranormal weighted shifts on directed trees and give some examples. Moreover we give some examples which show that the spectrum of a positive absolutely norm attaining operator…

Let $\mathscr T=(V, \mathcal E)$ be a leafless, locally finite rooted directed tree. We associate with $\mathscr T$ a one parameter family of Dirichlet spaces $\mathscr H_q~(q \geqslant 1)$, which turn out to be Hilbert spaces of…

In this paper, we introduce a discrete analogue of weighted Hardy spaces on rooted trees and study weighted composition operators between them in detail. In particular, we characterize bounded and compact weighted composition operators…

We will define an operad $\mathcal{B}^0$ on planar rooted trees. $\mathcal{B}^0$ is analgous to the $NAP$-operad in the non-planar tree setting. We will define a family of "current-preserving" operads $\mathcal{B}^\lambda$ depending on a…

In this paper, we define a new kind of weighted tree automata where the weights are only supported by final states. We show that these automata are sequentializable and we study their closures under classical regular and algebraic…

We study operators in four-dimensional gauge theories which are localized on a straight line, create electric and magnetic flux, and in the UV limit break the conformal invariance in the minimal possible way. We call them Wilson-'t Hooft…

We study a family of tree-type diagrams that arise in studies of the cumulant expansion in discrete Erd\H os-R\'enyi random matrix models. Using a version of the Pr\" ufer code, we obtain an explicit expression for the number of tree-type…