Questions on aperiodicity in graphs

Question

Background

I am trying to understand different notions of aperiodicity in graphs, and how these concepts relate to aperiodic tilings. There is a different description of aperiodicity in graphs on this page. It says the following:

A directed graph is said to be aperiodic if there is no integer k > 1 that divides the length of every cycle of the graph. Equivalently, a graph is aperiodic if the greatest common divisor of the lengths of its cycles is one; this greatest common divisor for a graph $G$ is called the period of $G$.

I think this doesn't have anything to do with the type of aperiodicity found in, for instance, the Penrose tilings. (Please correct me if I'm wrong.) Let's call this definition (A) of graph aperiodicity, and leave it for now.

There's another definition of both non-periodicity and aperiodicity in graphs in the following MO question by Jesús Alvarez:

The concepts of being non-periodic and aperiodic for tilings have obvious versions for connected graphs with a countable set of vertices and a finite number of edges meeting at each vertex. A graph $G$ of this class is non-periodic when its group of graph isomorphisms is trivial (this is the group of isometries if the graph is considered as a metric space in the usual way). $G$ is aperiodic if its hull consists of non-periodic graphs. Here, the hull of $G$ consists of all graphs of this class that can be expressed as an increasing union of balls with the same center and increasing radius, each of them isometric to some ball in $G$ (using the metric structure).

Let's call these definitions (B) of non-periodicity and aperiodicity, respectively. I think I understand definition B of non-periodicity, because the idea of graph isomorphisms is explained quite well in this wiki entry.

However, I have difficulties understanding the concept of aperiodicity in this case. I understand the notion of a convex hull in Euclidean space, but for graphs it is still lacking. I will ask a set of questions on it at the bottom of the page.

Finally, there is a third definition, definition (C), of aperiodicity of graphs in the following presentation, also made by dr. Alvarez. This time, the definition pertains to graph colorings. It is as follows:

Let $G$ be a simple graph. A coloring $\phi$ on $G$ is said to be aperiodic or distinguishing if there are no non-trivial automorphisms of $G$ preserving $\phi$.

Notice that in this case, the definition of non-periodicity of a graph coloring is missing.

Questions

I have the following questions on these definitions:

1. With regards to the notion of aperiodicity described in definition (B): can the increasing union of balls taken from any graph node as its center? Or do we need to pick a specific one?
2. Can you provide an instructive example and a counterexample that elucidates definition (B) of aperiodicity in graphs?
3. Is there also a notion of non-periodicity in the context of the definition of aperiodicity in definition (C)? If so, what is it?
4. How do definitions (B) and (C) of graph (coloring) aperiodicity compare to one another?
5. What does, say, the P1 Penrose tiling look like in the context of definitions B and C of graph aperiodicity?