A uniform 6-regular graph consisting of triangles and quadrilaterals

Question

I am interested in a special uniform graph that can be constructed by attaching succesively three triangles to each other such that always four of them form a circle. (The nodes of my graph are the points where the triangles meet.)

Its motivation is a simplified friendship graph: consider a group of people of which each has six friends, which are half-pairwise friends to each other.

The graph (when extended to infinity) is $6$-regular and each node has exactly 21 neighbours at graph distance $2$. I guess it's not the only one that has this property, but assumably it is the most regular one (in fact it is completely symmetric, isn't it?) In a sense it's also the most "clustered".

My question is threefold:

• Has someone seen this graph in its whole fractal beauty?

• Under what name is this graph known?

• How do I calculate the adjacency matrix of this graph (i.e. a finite portion of it)?

Something like $a_{ij} = 1$ iff $\Phi(i,j)$ with an explicit expression $\Phi(i,j)$ would be welcome.

Answer 1

Your graph can be nicely embedded in the hyperbolic plane as the alternated octagonal tiling, with three triangles and three squares meeting at each vertex.

(Why "octagonal"? Because, as a graph, it is the half-square of the octagonal tiling where three octagons meet at each vertex. To put it differently: starting from the octagonal tiling, if you replace every other vertex by a triangle, and grow these triangles until their corners touch, you get the alternated octagonal tiling.)

As far as seeing it in its whole fractal beauty, there's M.C. Escher's Circle Limit III:

Answer 2

Tegula (thanks to user Jaap!) gave me this tiling:

It's the first when you filter by geometry = hyperbolic and vertex degree = 6 and the second when you filter by number of non-equivalent tiles = 2, number of non-equivalent edges = 1, number of non-equivalent vertices = 1.

[Side question: How does Tegula's nomenclature n:3 t:2 e:1 v:1 g:*433 relate to the vertex configuration 3.4.3.4.3.4? What especially does n:3 mean?]

Answer 3

Up to a not too large diameter the tritetragonal tiling can be quite easily drawn in the Euclidean plane and gives an idea of its fractal nature. What's more important: It allows to count vertices at distance 3 and 4, and all in all: to better understand and analyze the graph visually, at least locally:

By the way, I have an adjacency matrix for this particular graph, and I have an idea how to get it for even larger diameters (step by step, not in general).

This is another – less geometrical, more graphical – view of the graph, its shape depending on the order in which the nodes were created:

For the sake of completeness the same with smaller diameter: