Related papers: Watching Systems in graphs: an extension of Identi…
The work in this article is concerned with two different types of families of finite sets: separating families and splitting families (they are also called "systems"). These families have applications in combinatorial search, coding theory,…
We give upper and lower bounds on the spectral radius of a graph in terms of the number of walks. We generalize a number of known results.
We use probabilistic methods to find lower bounds on the maximum number, in a graph with domination number \gamma, of dominating sets of size \gamma. We find that we can randomly generate a graph that, w.h.p., is dominated by almost all…
We introduce graphcodes, a novel multi-scale summary of the topological properties of a dataset that is based on the well-established theory of persistent homology. Graphcodes handle datasets that are filtered along two real-valued scale…
In the literature, several identification problems in graphs have been studied, of which, the most widely studied are the ones based on dominating sets as a tool of identification. Hereby, the objective is to separate any two vertices of a…
A graph is a mathematical object consisting of a set of vertices and a set of edges connecting vertices. Graphs can be drawn on paper in various ways, but until recently all published methods of drawing graphs have had undesirable…
In this paper, we study the complexity of the edge monitoring problem. A vertex $v$ monitors an edge $e$ if both extremities together with $v$ form a triangle in the graph. Given a graph $G=(V,E)$ and a weight function on edges $c$ where…
An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and unique. On a finite graph, the density of a code is…
Introduced the quantitative measure of the structural complexity of the graph (complex network, etc.) based on a procedure similar to the renormalization process, considering the difference between actual and averaged graph structures on…
Topological mapping of a large physical system on a graph, and its decomposition using universal measures is proposed. We find inherent limits to the potential for optimization of a given system and its approximate representations by…
Compound graphs are networks in which vertices can be grouped into larger subsets, with these subsets capable of further grouping, resulting in a nesting that can be many levels deep. In several applications, including biological workflows,…
In this paper we generalise the notion of visibility from a point in an integer lattice to the setting of graph theory. For a vertex $x$ of a connected graph $G$, we say that a set $S \subseteq V(G)$ is an \emph{$x$-position set} if for any…
Link prediction systems (e.g. recommender systems) typically use graph topology as one of their main sources of information. However, automorphisms and related properties of graphs beget inherent limits in predictability. We calculate hard…
Some of the basic properties of any dynamical system can be summarized by a graph. The dynamical systems in our theory run from maps like the logistic map to ordinary differential equations to dissipative partial differential equations. Our…
We consider the notion of herdability, a set-based reachability condition, which asks whether the state of a system can be controlled to be element-wise larger than a non-negative threshold. First a number of foundational results on…
Graphs are typically visualized as node-link diagrams. Although there is a fair amount of research focusing on crossing minimization to improve readability, little attention has been paid on how to handle crossings when they are an…
Mutual visibility in graphs provides a framework for analysing how vertices can observe one another along shortest paths free of internal obstructions. The visibility polynomial, which enumerates mutual-visibility sets of all orders, has…
Monitoring edge-geodetic sets in a graph are subsets of vertices such that every edge of the graph must lie on all the shortest paths between two vertices of the monitoring set. These objects were introduced in a work by Foucaud, Krishna…
Minimal input/output selection is investigated in this paper for each subsystem of a networked system. Some novel sufficient conditions are derived respectively for the controllability and observability of a networked system, as well as…
Among all characteristics exhibited by natural and man-made networks the small-world phenomenon is surely the most relevant and popular. But despite its significance, a reliable and comparable quantification of the question `how small is a…