Related papers: Two notions of unit distance graphs
The Fr\'echet distance is a popular distance measure between trajectories or curves in space, or between walks in graphs. We study computing the Fr\'echet distance between walks in the $d$-dimensional grid graphs, i.e. $\mathbb{Z}^d$ where…
A graph $\Gamma$ is said to be distance-balanced if for any edge $uv$ of $\Gamma$, the number of vertices closer to $u$ than to $v$ is equal to the number of vertices closer to $v$ than to $u$, and it is called nicely distance-balanced if…
We study the concept of the continuous mean distance of a weighted graph. For connected unweighted graphs, the mean distance can be defined as the arithmetic mean of the distances between all pairs of vertices. This parameter provides a…
In this note we give a construction proving that the Gray graph, which is the smallest cubic semi-symmetric graph, is a unit-distance graph.
A set $W\subseteq V(G)$ is called a resolving set, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. A resolving set for $G$…
We study unit disk visibility graphs, where the visibility relation between a pair of geometric entities is defined by not only obstacles, but also the distance between them. That is, two entities are not mutually visible if they are too…
The unit Euclidean distance degree and the generic Euclidean distance degree are two well-studied invariants of projective varieties. These quantities measure the algebraic complexity of nearest-point problems on a variety, and in many…
We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit…
Let $\ell$ denote a positive integer. A connected graph $\G$ of diameter at least $\ell$ is said to be $\ell${\it -distance-balanced} whenever for any pair of vertices $u,v$ of $\G$ such that $d(u,v)=\ell$, the number of vertices closer to…
In this paper we study the distance Ramsey number $R_{{\it D}}(s,t,d)$. The \textit{distance Ramsey number} $R_{{\it D}}(s,t,d) $ is the minimum number $n$ such that for any graph $ G $ on $ n $ vertices, either $G$ contains an induced $ s…
In this note we present eleven unit distance embeddings of the Heawood graph, i.e. the point-line incidence graph of the finite projective plane of order two, by way of pictures and 15 digit approximations of the coordinates of the…
An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth…
In this paper, we present a new metric distance for comparing two large graphs to find similarities and differences between them based on one of the most important graph structural properties, which is Node Adjacency Information, for all…
We give an overview of different approaches to measuring the similarity of, or the distance between, two graphs, highlighting connections between these approaches. We also discuss the complexity of computing the distances.
We present a framework for embedding graph structured data into a vector space, taking into account node features and topology of a graph into the optimal transport (OT) problem. Then we propose a novel distance between two graphs, named…
We introduce a new graph-theoretic concept in the area of network monitoring. A set $M$ of vertices of a graph $G$ is a \emph{distance-edge-monitoring set} if for every edge $e$ of $G$, there is a vertex $x$ of $M$ and a vertex $y$ of $G$…
Let G be a simple, connected graph on n vertices. Let d_G(u,v) denote the distance between vertices u and v in G. A subgraph H of G is isometric if d_H(u,v)=d_G(u,v) for every u,v in V(H). We say that G is a distance-preserving graph if G…
For a graph $\Gamma$, the {\em distance} $d_\Gamma(u,v)$ between two distinct vertices $u$ and $v$ in $\Gamma$ is defined as the length of the shortest path from $u$ to $v$, and the {\em diameter} $\mathrm{diam}(\Gamma)$ of $\Gamma$ is the…
Temporal graphs are commonly used to represent time-resolved relations between entities in many natural and artificial systems. Many techniques were devised to investigate the evolution of temporal graphs by comparing their state at…
In this paper we propose and study a new structural invariant for graphs, called distance-unbalanced\-ness, as a measure of how much a graph is (un)balanced in terms of distances. Explicit formulas are presented for several classes of…