Combinatorics and Graph Theory

2406 Submissions

[2] viXra:2406.0086 [pdf] submitted on 2024-06-18 20:53:41

A New Upper Bound for the Heilbronn Triangle Problem

Authors: Theophilus Agama
Comments: 10 Pages.

Using ideas from the geometry of compression, we improve on the current upper bound of Heilbronn's triangle problem. In particular, by letting $Delta(s)$ denotes the minimal area of the triangle induced by $s$ points in a unit disc, then we have the upper bound $$Delta(s)ll frac{1}{s^{frac{3}{2}-epsilon}}$$ for small $epsilon:=epsilon(s)>0$.
Category: Combinatorics and Graph Theory

[1] viXra:2406.0027 [pdf] submitted on 2024-06-06 00:49:05

Graphs and Their Symmetries

Authors: Teo Banica
Comments: 400 Pages.

This is an introduction to graph theory, from a geometric viewpoint. A finite graph $X$ is described by its adjacency matrix $din M_N(0,1)$, which can be thought of as a kind of discrete Laplacian, and we first discuss the basics of graph theory, by using $d$ and linear algebra tools. Then we discuss the computation of the classical and quantum symmetry groups $G(X)subset G^+(X)$, which must leave invariant the eigenspaces of $d$. Finally, we discuss similar questions for the quantum graphs, with these being again described by certain matrices $din M_N(mathbb C)$, but in a more twisted way.
Category: Combinatorics and Graph Theory