Yeah, do we have some papers on that?

Maybe this book will help. Follow the link! There are a lot of references available on net!

http://books.google.rs/books?id=xg2fBfXUtGgC&printsec=frontcover&dq=editions:4IthdPFpuvIC&hl=en&sa=X&ei=sdt7UuanKsaU4ATIzYDgCw&redir_esc=y#v=onepage&q&f=false

Ljubomir,

Great post! Many thanks for posting the book. It is very interesting.

@James, I am glad that I could help. Thanks.

@Vitaly,if You look into a contents, You will find parts about Graphs.See by Google Books.

Regards,

Ljubomir

@Ljubomir, Thank you . I think this topic is interesting to work on..

@Ljubomir But the books, I think there is nothing regarding Graph theory. I need some graph theory related books

When it comes to Graph Theory. What exactly are you looking for? This type of fractal is a geometric structure that can be treated more like a graph embedding (drawing) if you want it in a more graph theoretic sense. I'd urge if you need more graph theory to understand say the book they gave (which I might say is an excellent source for geometry, graphics, and structure among many other thing) to maybe invest in learning a bit more to see how it fits into the mix of graph drawing, fractal geometry, topology, graphics, and more! If you want a book, this page http://www.arsetmathesis.nl/bruno0402.htm (you will need to translate it) talks a lot about this construction, and gives one book "The wondrous exploration field of plane geometry" by Bruno Ernst. I'd urge you to search that as the reference for more recent citations to see where it is being used, then use that as keywords to find more sources, or even more books.

@ Daniel, I want to check the graph theoretic properties, just like regularity, geodeticity, and boundary value problems etc. the distances I hope will be interesting to study

Have you explored the literature on this fractal? If you don't find anything, it may be worthwhile to study its structure. For example, I believe if you were to look at the graph that you could gather from the embedding that it is outerplanar, for example. I say this because the faces are only squares and triangles and stem off eachother. If that is true (prove it first... or cite a paper which has done such if it is true), gives you a lot of lovely properties that other graphs will not have.

For example, you can colour any simple outerplanar graph in polynomial time. This is because its chromatic number would be at most three! That being said, tinker with things yourself and see if it is interesting based on what has been done with them.

Could anybody recall what a Pythagorean graph is? Connect two integers if they belong to the same pythagorean triple? So its an infinite vertex set graph, arguably a remote province of graph theory...

@Vitaly: it seems its not the definition i said!

If you take the definition like in http://www.ijser.org/paper/A-Note-on-Cordial-Edge-Cordial-Labeling-of-Pythagoras-Tree-Fractal-Graphs.html then the dual is easy to characterize as the dual of a tree which is the universal cover of (3,4) bipartite biregular graphs. It is also easy to generalize to universal covers of (k,l) graphs.

See http://ac.els-cdn.com/S0195669896900402/1-s2.0-S0195669896900402-main.pdf?_tid=c1130664-4aeb-11e3-8735-00000aab0f27&acdnat=1384186397_4312a23a4d3309c054079f0cf2ff4887

for a spectral theory of this graph, using orthogonal polynomials