2 Some basic notions 2.5 Graphs as metric spaces 3 Eulerian and Hamiltonian cycles

2.6 Some questions

Question 2.47.

Instead of counting $|B_{n}(0)|$ , consider the following counting. Let $f_{d}(r)=|\{(n_{1},\ldots,n_{d})\in\mathbb{Z}^{d}:\sum_{i=1}^{d}n_{i}^{2}\leq r% ^{2}:\}|$ be the number of lattice points within $r$ radius ball in $\mathbb{R}^{d}$ . Show that $f_{d}(r)/r^{d}\to$ a constant. To determine exact asymptotics of $f_{2}(r)$ is known as the Gauss circle problem.

Here are two questions from geometric group theory which can be stated in the language of graph theory. For more on this fascinating subject, refer to [Clay and Margalit 2017].

Question 2.48.

Suppose $G$ is the Cayley graph of a finitely generated countable group such that $n^{-d}|B_{n}(e)|\to\infty$ for all $d\in(0,\infty)$ . Is it true that $n^{-a}\log|B_{n}(e)|\to\infty$ for some $a>0$ ?

Question 2.49.

Can you construct a finitely generated group such that its Cayley graph has the following growth property for some $a<0.7$ ?

0<\liminf_{n\to\infty}n^{-a}\log|B_{n}(e)|\leq\limsup_{n\to\infty}n^{-a}\log|B% _{n}(e)|<\infty.

Now some questions on self-avoiding walks.

Question 2.50.

Can you calculate $\kappa_{d}$ for any $d\geq 2$ ?

Note : If you have solutions for any of the above four questions, please consider submitting them here. It has been shown that $\kappa=\sqrt{2+\sqrt{2}}$ for the hexagonal lattice (see Figure 2.6) using discrete complex analysis [Duminil-Copin and Smirnov 2012]. In listing problems in IMO which can lead to research problems, Stanislav Smirnov mentions this ([Smirnov 2011]).

Refer to caption — Figure 2.6: Hexagonal lattice

2.6 ***Some questions***

Question 2.47.

Question 2.48.

Question 2.49.

Question 2.50.

2.6 Some questions