Related papers: Graphs related to $2$-dimensional simplex codes

Let \( D \) be a strongly connected digraph. The average distance of a vertex \( v \) in \( D \) is defined as the arithmetic mean of the distances from \( v \) to all other vertices in \( D \). The remoteness \( \rho(D) \) of \( D \) is…

For any finite, simple graph $G = (V,E)$, its $2$-distance graph $G_2$ is a graph having the same vertex set $V$ where two vertices are adjacent if and only if their distance is $2$ in $G$. Connectivity and diameter properties of these…

We show that large subsets of vector spaces over finite fields determine certain point configurations with prescribed distance structure. More specifically, we consider the complete graph with vertices as the points of $A \subseteq…

We consider the Cayley graph ${\rm C}(W,S)$ of a Coxeter system $(W,S)$ and describe all maximal $2$-cliques in this graph, i.e. maximal subsets in the vertex set such that the distance between any two distinct elements is equal to $2$. As…

Let $G$ be a connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity and remoteness of $G$ are defined as the minimum and maximum,…

In this paper we systematically study various properties of the distance graph in ${\Bbb F}_q^d$, the $d$-dimensional vector space over the finite field ${\Bbb F}_q$ with $q$ elements. In the process we compute the diameter of distance…

Let $n=2^k-1$ and $m=2^{k-2}$ for a certain $k\ge 3$. Consider the point-line geometry of $2m$-element subsets of an $n$-element set. Maximal singular subspaces of this geometry correspond to binary simplex codes of dimension $k$. For $k\ge…

In this paper, we study the problem that which of distance-regular graphs admit a perfect $1$-code. Among other results, we characterize distance-regular line graphs which admit a perfect $1$-code. Moreover, we characterize all known…

Let $G$ be a connected graph with $V(G)=\{v_1,\ldots,v_n\}$. The $(i,j)$-entry of the distance matrix $D(G)$ of $G$ is the distance between $v_i$ and $v_j$. In this article, using the well-known Ramsey's theorem, we prove that for each…

The characterization of distance-regular Cayley graphs originated from the problem of identifying strongly regular Cayley graphs, or equivalently, regular partial difference sets. In this paper, a classification of distance-regular Cayley…

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…

The distance matrix of a connected graph is the symmetric matrix with columns and rows indexed by the vertices and entries that are the pairwise distances between the corresponding vertices. We give a construction for graphs which differ in…

The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [4] by introducing a vast hierarchy of…

The simplex graph $S(G)$ of a graph $G$ is defined as the graph whose vertices are the cliques of $G$ (including the empty set), with two vertices being adjacent if, as cliques of $G$, they differ in exactly one vertex. Simplex graphs form…

Graphs on integer points of polytopes whose edges come from a set of allowed differences are studied. It is shown that any simple graph can be embedded in that way. The minimal dimension of such a representation is the fiber dimension of…

Let $G$ be a finite, connected graph. The average distance of a vertex $v$ of $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The remoteness $\rho(G)$ and the proximity $\pi(G)$ of $G$ are the maximum and…

We revisit a new type of a Voronoi diagram, in which distance is measured from a point to a pair of points. We consider a few more such distance functions, based on geometric primitives, and analyze the structure and complexity of the…

The unit distance graph $G_{\mathbb{R}^d}^1$ is the infinite graph whose nodes are points in $\mathbb{R}^d$, with an edge between two points if the Euclidean distance between these points is 1. The 2-dimensional version $G_{\mathbb{R}^2}^1$…

A resolving set of a graph is a set of vertices with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. In this paper, we construct a resolving set of Johnson graphs, doubled Odd…

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.