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Say that a graph $G$ is emph{representable in $R ^n$} if there is a map $f$ from its vertex set into the Euclidean space $R ^n$ such that $| f(x) - f(x)| = | f(y) - f(y)|$ iff ${x,x}$ and ${y, y}$ are both edges or both non-edges in $G$. The purpose of this note is to present the proof of the following result, due to Einhorn and Schoenberg: if $G$ finite is neither complete nor independent, then it is representable in $R ^{|G|-2}$. A similar result also holds in the case of finite complete edge-colored graphs.
We consider finite simple graphs. Given a graph $H$ and a positive integer $n,$ the Tur{a}n number of $H$ for the order $n,$ denoted ${rm ex}(n,H),$ is the maximum size of a graph of order $n$ not containing $H$ as a subgraph. ErdH{o}s posed the foll
In this paper we define the notion of monic representation for the $C^*$-algebras of finite higher-rank graphs with no sources, and undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the co
The inverse eigenvalue problem of a graph $G$ aims to find all possible spectra for matrices whose $(i,j)$-entry, for $i eq j$, is nonzero precisely when $i$ is adjacent to $j$. In this work, the inverse eigenvalue problem is completely solved for a
We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep connections to
We analyze the behavior of the Euclidean algorithm applied to pairs (g,f) of univariate nonconstant polynomials over a finite field F_q of q elements when the highest-degree polynomial g is fixed. Considering all the elements f of fixed degree, we es