Sets of Complex Unit Vectors with Two Angles and Distance-Regular Graphs

A lower bound for 4-regular planar unit distance graphs

We perform an exhaustive search for the minimum 4-regular unit distance graph resulting in a lower bound of 34 vertices.

Rational Distances with Rational Angles

In 1946 Erd\H os asked for the maximum number of unit distances, $u(n)$, among $n$ points in the plane. He showed that $u(n)> n^{1+c/\log\log n}$ and conjectured that this was the true magnitude. The best known upper bound is $u(n)<cn^{4/3}$, due to Spencer, Szemerédi and Trotter. We show that the upper bound $n^{1+6/\sqrt{\log n}}$ holds if we only consider unit distances with rational angle, by which we mean that the line through the pair of points makes a rational angle in degrees with the x-axis. Using an algebraic theorem of Mann we get a uniform bound on the number of paths between two f…

酉Cayley图与整正则图上的Grover游走

Grover walks on unitary Cayley graphs and integral regular graphs

本文研究酉Cayley图与整正则图上的Grover游走，刻画了周期酉Cayley图，给出循环图完美态传输充要条件，证明仅4个酉Cayley图有完美态传输，还给出整正则图上Grover游走周期性的谱刻画。

4-regular planar unit triangle graphs without additional triangles

In this article we proof the existence of 4-regular planar unit-distance graphs consisting only of unit triangles without additional triangles. It is shown that the smallest number of unit triangles is $\leq$6422.