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It is a well-known fact that a graph of diameter $d$ has at least $d+1$ eigenvalues. Let us call a graph \emph{$d$-extremal} if it has diameter $d$ and exactly $d+1$ eigenvalues. Such graphs have been intensively studied by various authors.…
For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),...,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where…
The mixed metric dimension ${\rm mdim}(G)$ of a graph $G$ is the cardinality of a smallest set of vertices that (metrically) resolves each pair of elements from $V(G)\cup E(G)$. We say that $G$ is a max-mdim graph if ${\rm mdim}(G) = n(G)$.…
Let $d, n \in \mathbb{Z}^+$ such that $1\leq d \leq n$. A $d$-code $\mathcal{C} \subset \mathbb{F}_q^{n \times n}$ is a subset of order $n$ square matrices with the property that for all pairs of distinct elements in $\mathcal{C}$, the rank…
A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was only partially resolved, but standard…
In 2008, Vallentin made a conjecture involving the least distortion of an embedding of a distance-regular graph into Euclidean space. Vallentin's conjecture implies that for a least distortion Euclidean embedding of a distance-regular graph…
Here we study the multipacking problems for geometric point sets with respect to their Euclidean distances. We consider a set of $n$ points $P$ and define $N_s[v]$ as the subset of $P$ that includes the $s$ nearest points of $v \in P$ and…
We analyze the distribution of the distance between two nodes, sampled uniformly at random, in digraphs generated via the directed configuration model, in the supercritical regime. Under the assumption that the covariance between the…
The introduction of geometry has proven instrumental in the efforts towards more realistic models for real-world networks. In Geometric Inhomogeneous Random Graphs (GIRGs), Euclidean Geometry induces clustering of the vertices, which is…
The diameter of a graph is the maximum distance among all pairs of vertices. Thus a graph $G$ has diameter $d$ if any two vertices are at distance at most $d$ and there are two vertices at distance $d$. We are interested in studying the…
Geometrical objects with integral side lengths have fascinated mathematicians through the ages. We call a set $P=\{p_1,...,p_n\}\subset\mathbb{Z}^2$ a maximal integral point set over $\mathbb{Z}^2$ if all pairwise distances are integral and…
A generator of a metric space is a set $S$ of points in the space with the property that every point of the space is uniquely determined by its distances from the elements of $S$. Given a simple graph $G=(V,E)$, we define the distance…
We review the nearly complete classification project for finite distance-transitive graphs and compile a list of all known graphs. Interestingly, we find that those graphs with diameter larger than 4, apart from a small finite number of…
We address an analog of a problem introduced by Erd\H{o}s and Fishburn, itself an inverse formulation of the famous Erd\H{o}s distance problem, in which the usual Euclidean distance is replaced with the metric induced by the $\ell^1$-norm,…
A graph is called a $k$-planar unit distance graph if it can be drawn in the plane such that every edge is a unit line segment and is involved in at most $k$ crossings. We investigate $u_k(n)$, the maximum number of edges of such graphs on…
For a fixed graph G, a maximal independent set is an independent set that is not a proper subset of any other independent set. P. Erd\"os, and independently, J. W. Moon and L. Moser, and R. E. Miller and D. E. Muller, determined the maximum…
In this work, the classical Nelson -- Hadwiger problem is studied which lies on the edge of combinatorial geometry and graph theory. It concerns colorings of distance graphs in $ {\mathbb R}^n $, i.e., graphs such that their vertices are…
We revisit the classical Unit Distance Problem posed by Erd\H{o}s in 1946. While the upper bound of $O(n^{4/3})$ established by Spencer, Szemer'edi, and Trotter (1984) is tight for systems of pseudo-circles, it fails to account for the…
In a graph $G$, a subset of vertices is a dissociation set if it induces a subgraph with vertex degree at most 1. A maximum dissociation set is a dissociation set of maximum cardinality. The dissociation number of $G$, denoted by $\psi(G)$,…
Let $n$ points be in crescent configurations in $\mathbb{R}^d$ if they lie in general position in $\mathbb{R}^d$ and determine $n-1$ distinct distances, such that for every $1 \leq i \leq n-1$ there is a distance that occurs exactly $i$…