Related papers: Aperiodicity Conditions in Topological $k$-Graphs
We prove that every 2k-edge-connected graph with countably many edge-ends admits a k-arc-connected orientation, extending the previous result by Assem, Koloschin and Pitz that also assumed the hypothesis of the graph being locally finite.…
A connected graph $G$ is said to be $k$-connected if it has more than $k$ vertices and remains connected whenever fewer than $k$ vertices are deleted. In this paper, for a connected graph $G$ with sufficiently large order, we present a…
In this paper, we develop twisted $K$-theory for stacks, where the twisted class is given by an $S^1$-gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure $K^i_\alpha…
We investigate existence and nonexistence of action ground states and nodal action ground states for the nonlinear Schr\"odinger equation on noncompact metric graphs with rather general boundary conditions. We first obtain abstract…
We exploit a result by Nerman which shows that conditional limit theorems hold when a certain monotonicity condition is satisfied. Our main result is an application to vertex degrees in random graphs, where we obtain asymptotic normality…
We investigate conditions on a graph $C^*$-algebra for the existence of a faithful semifinite trace. Using such a trace and the natural gauge action of the circle on the graph algebra, we construct a smooth $(1,\infty)$-summable semfinite…
We show that for an associative algebra A and its ideal I such that the I-adic topology on A coincides with the p-adic topology, the relative continuous K-theory pro-spectrum "lim"K(A_i, IA_i), where A_i :=A/p^i A, is naturally isogenous to…
The generalized Markoff mod $p$ graph is defined via the equation $x^2+y^2+z^2=xyz+\kappa$ over the finite field $\mathbb{F}_p$ of prime order $p$. In this paper, we investigate the topological properties of the graph such as non-planarity,…
We study the version of the $k$-disjoint paths problem where $k$ demand pairs $(s_1,t_1)$, $\dots$, $(s_k,t_k)$ are specified in the input and the paths in the solution are allowed to intersect, but such that no vertex is on more than $c$…
We show that if $E$ is an arbitrary acyclic graph then the Leavitt path algebra $L_K(E)$ is locally $K$-matricial; that is, $L_K(E)$ is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the…
Algebraic curves have a discrete analogue in finite graphs. Pursuing this analogy we prove a Torelli theorem for graphs. Namely, we show that two graphs have the same Albanese torus if and only if the graphs obtained from them by…
Network reliability is a well-studied problem that requires to measure the probability that a target node is reachable from a source node in a probabilistic (or uncertain) graph, i.e., a graph where every edge is assigned a probability of…
We examine $t$-colourings of oriented graphs in which, for a fixed integer $k \geq 1$, vertices joined by a directed path of length at most $k$ must be assigned different colours. A homomorphism model that extends the ideas of Sherk for the…
Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological…
We take a general approach to uncertainty on product spaces, and give sufficient conditions for the independence structures of uncertainty measures to satisfy graphoid properties. Since these conditions are arguably more intuitive than some…
Infinite graphs are finitary in the sense that their points are connected via finite paths. So what would an infinitary generalization of finite graphs look like? Usually this question is answered with the aid of topology, e.g. in the case…
We prove that the C*-algebra of a directed graph $E$ is liminal iff the graph satisfies the finiteness condition: if $p$ is an infinite path or a path ending with a sink or an infinite emitter, and if $v$ is any vertex, then there are only…
Let $G=(V(G), E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For $k\geq2$ and given any subset $S\subseteq|V(G)|$ with $|S|=k$, if a graph $G$ of order $|V(G)|\geq k+1$ always has a spanning tree $T$ such that $S$ is…
This introduction to graphs and graph algebras provides the optimal bound for the number of all paths of length $k$ in a graph with $N\geq k$ edges and no loops. Our proof relies on a construction of a number of terminating algorithms that…
This paper develops a structural theory of unique shortest paths in real-weighted graphs. Our main goal is to characterize exactly which sets of node sequences, which we call path systems, can be realized as unique shortest paths in a graph…