Browsing by Subject "weighted graph"

Now showing 1 - 3 of 3

Access status: Open Access ,
A sampling theory for infinite weighted graphs
(2011) Jørgensen, Palle E.T.
We prove two sampling theorems for infinite (countable discrete) weighted graphs $G$, one example being »large grids of resistors« i.e., networks and systems of resistors. We show that there is natural ambient continuum X containing $G$, and there are Hilbert spaces of functions on $X$ that allow interpolation by sampling values of the functions restricted only on the vertices in $G$. We sample functions on $X$ from their discrete values picked in the vertex-subset $G$. We prove two theorems that allow for such realistic ambient spaces $X$ for a fixed graph $G$, and for interpolation kernels in function Hilbert spaces on $X$, sampling only from points in the subset of vertices in $G.$ A continuum is often not apparent at the outset from the given graph $G$. We will solve this problem with the use of ideas from stochastic integration.
Access status: Open Access ,
Frames and factorization of graph Laplacians
(2015) Jørgensen, Palle E.T.; Tian, Feng
Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space $\mathscr{H}_{E}$ of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in $\mathscr{H}_{E}$ we characterize the Friedrichs extension of the $\mathscr{H}_{E}$-graph Laplacian. We consider infinite connected network-graphs $G=(V,E)$, $V$ for vertices, and $E$ for edges. To every conductance function $c$ on the edges $E$ of $G$, there is an associated pair ($\mathscr{H}_{E}$, $\Delta$) where $\mathscr{H}_{E}$ in an energy Hilbert space, and $\Delta\left(=\Delta_{c}\right)$ is the $c$-graph Laplacian; both depending on the choice of conductance function $c$. When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in $\mathscr{H}_{E}$ consisting of dipoles. Now $\Delta$ is a well-defined semibounded Hermitian operator in both of the Hilbert $l^{2}\left(V\right)$ and $\mathscr{H}_{E}$. It is known to automatically be essentially selfadjoint as an $l^{2}\left(V\right)$-operator, but generally not as an $\mathscr{H}_{E}$ operator. Hence as an $\mathscr{H}_{E}$ operator it has a Friedrichs extension. In this paper we offer two results for the Friedrichs extension: a characterization and a factorization. The latter is via $l^{2}\left(V\right)$.
Access status: Open Access ,
Lightweight paths in graphs
(Wydawnictwa AGH, 2019) Harant, Jochen; Jendroľ, Stanislav
Let $k$ be a positive integer, $G$ be a graph on $V(G)$ containing a path on $k$ vertices, and w be a weight function assigning each vertex $v \in V(G)$ a real weight $w(v)$. Upper bounds on the weight $w(P)=\sum_{v\in V(P)}w(v)$ of $P$ are presented, where $P$ is chosen among all paths of $G$ on $k$ vertices with smallest weight.