Related papers: The mixed metric dimension of flower snarks and wh…
The \emph{metric dimension} of a graph $G$, denoted by $\dim(G)$, is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let $G_1$ and $G_2$ be disjoint copies of a graph $G$…
The cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. For certain monoids connected with combinatorics, such as the plactic monoid (the monoid…
The metric dimension of a graph is the minimum size of a set of vertices such that each vertex is uniquely determined by the distances to the vertices of that set. Our aim is to upper-bound the order $n$ of a graph in terms of its diameter…
In this manuscript, we provide a concise review of the concept of metric dimension for both deterministic as well as random graphs. Algorithms to approximate this quantity, as well as potential applications, are also reviewed. This work has…
A {\em snark} is a cubic cyclically 4-edge connected graph with edge chromatic number four and girth at least five. We say that a graph $G$ is {\em odd 2-factored} if for each 2-factor F of G each cycle of F is odd. In this paper, we…
The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner…
We investigate how effectively finite metric spaces can be distinguished by distance-based invariants. As model spaces, we consider regular polygons, cycle graphs, and their generalization, circular metric spaces, and as invariants we…
A {\em flower} is a coin graph representation of the wheel graph. A {\em petal} of the wheel graph is an edge to the center vertex. In this paper we investigate flowers whose coins have integer radii. For an $n$-petaled flower we show there…
In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) (resp. E(G)) is called the vertex (resp. edge) metric dimension of G. In [16] it was shown that both vertex and edge metric…
The Metric Dimension problem asks for a minimum-sized resolving set in a given (unweighted, undirected) graph $G$. Here, a set $S \subseteq V(G)$ is resolving if no two distinct vertices of $G$ have the same distance vector to $S$. The…
We study a 5d gravity theory with a warped metric and show that two N = 2 supersymmetric quantum-mechanical systems are hidden in the 4d spectrum. The supersymmetry can be regarded as a remnant of higher-dimensional general coordinate…
A {\em resolving set} for a graph $\Gamma$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The {\em metric dimension} of $\Gamma$ is the…
Radial Moore graphs and digraphs are extremal graphs related to the Moore ones where the distance-preserving spanning tree is preserved for some vertices. This leads to classify them according to their proximity to being a Moore graph or…
A wheel is a graph that consists of a chordless cycle of length at least 4 plus a vertex with at least three neighbors on the cycle. It was shown recently that detecting induced wheels is an NP-complete problem. In contrast, it is shown…
A set of vertices $S$ resolves a graph if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of a graph is the minimum cardinality of a resolving set of the graph. Fix a connected…
The {\em metric dimension} of a graph $\Gamma$ is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. We consider the Grassmann graph…
The Johnson graph J(n,N) is defined as the graph whose vertices are the n-subsets of the set {1,2,...,N}, where two vertices are adjacent if they share exactly n - 1 elements. Unlike Johnson graphs, induced subgraphs of Johnson graphs (JIS…
We define a complete invariant for doodles on a 2-sphere which takes values in series of chord diagrams of certain type. The coefficients at the diagrams with $n$ chords are finite type invariants of doodles of order at most $2n$.
Mixed anomalies, higher form symmetries, two-group symmetries and non-invertible symmetries have proved to be useful in providing non-trivial constraints on the dynamics of quantum field theories. We study mixed anomalies involving discrete…
For an ordered set W = {w1,w2,...,wk} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W) := (d(v,w1),d(v,w2),...,d(v,wk)) is called the (metric) representation of v with respect to W, where d(x,y) is the distance…