Since manuscript \cite{R}, recently posted on arXiv, is closely related to our previous work, we would like to clarify some facts about it. Namely, we would like to point out that several results in \cite{R} are already known (Theorem 1 and Corollaries 2, 3 and 4) and some observations regarding our work \cite{FJ} are not correct. We also notice that R\'{e}dei's Theorem, a classical result in graph theory, gives a simple tool to show Theorems 5 and 9.
We introduce and study a finite simple graph of algebraic origin: the Vieta graph on the solution set over $\mathbb{F}_p$ to a symmetric, multivariate equation which is quadratic in each variable. This construction is a broad generalization of the Markoff graph over $\mathbb{F}_p$, extensively studied in the recent lit...
In 1999, two papers of Benjamini, Lyons, Peres, and Schramm showed that for Bernoulli percolation on unimodular nonamenable quasi-transitive graphs, there are no infinite clusters at criticality. Using the theory of countable Borel equivalence relations and Gaboriau's cluster graphing construction, we give a short proo...
In 1943, Hadwiger formulated his celebrated conjecture, connecting the chromatic number $\chi(G)$ of a finite, simple, undirected graph with the cardinality of the largest complete minor, $\eta(G)$. The disjoint union of all finite complete graphs shows that Hadwiger's conjecture fails for infinite, but a slightly weak...
For a graph $G$, let $\alpha(G)$ be the second smallest eigenvalue of the Laplacian matrix of $G$, also known as the algebraic connectivity. Algebraic connectivity plays an important role in characterizing the connectivity of graphs and convergence properties of networks. Kolokolnikov conjectured that among all graphs...
In these notes, we introduce the 50-year-old $K(\pi, 1)$ conjecture alongside Coxeter and Artin groups. Roughly speaking, the conjecture states that the complement in $\mathbb{C}^n$ of a"symmetric"configuration of hyperplanes is a $K(\pi, 1)$ space. Our end goal is to present a proof of the conjecture in the so-called...
Giovanni Paolini· Winter Braids Lecture Notes· 0 citations
We give a detailed account of the recent work of Hickman and Srivastava on counting integral points in a $ \delta $-neighbourhood of a dilate $ q \mathcal{M} $ of a non-degenerate curve $ \mathcal{M} \subset \mathbb{R}^{n} $, $ n \ge 3 $. Our aim is expository: we simplify the proof, provide additional details, refine...
Tian-Liang Wu· 0 citations
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