Any textbook talking about simplicial homology. For instance chapter 2 of Hatcher book http://www.math.cornell.edu/~hatcher/AT/ATpage.html. Simplicial decomposition give rise to special graphs in algebraic topology.

You can study different topology on graphs also. You can associate graph to topologic space (skeletal structure). Each time, you have numerous books on the subject. Try to make your question more accurate, please.

Many Thanks Dr. Olivier Ruatta for your answer and suggestions. More precisely I want to known :

Consider a set X={a,b,c,d} endowed with topology u={{\empty set}, X, {a}, {a,b}}, can we associate to the u graph?, we can consider on X several topologies, are there theorems about connections of their topologies with associated graphs?

Ok, it more about topology over graphs. But here, the topology does not change the graph. Are you looking for the ordered lattice associated to your topology ?

there is a strong relation between finite topologies and graphs (between orders and topologies too), for example, between scott topologies and partial orders (and the associated Hasse diagrams). In your case, your topology defines a

total preorder c=d

Dear Dr. Asier Estevan Muguerza, Thank you for your helpful suggestion, it seem be more relative to my question. Unfortunately, I don't found capable literature as monograph for this approach. Please inform me if you will find such kind of reference.

you can read somthingh about the topic in

-Bridges, D. S. and Mehta, G. B. (1995).

"Representations of preference orderings" Springer-Verlag, Berlin.

-Davey B.A. and Priestley H.A. "Introduction to lattices and order"

I will write you if I find a more interesting reference.

Thanks again Dr. Asier Estevan.

A modern approach about this topic can be found in:

- Computational topology by Harer and Edelsbrunner

persistent topology is an active area and graph plays an important role.

A nice classical introduction is in Algebraic Topology by Fulton.

Dear Dr. Luis Solarte, thank you for your answer. Approximately 10 years ago I studied algebraic topology, by my experience in this direction I have been deduced that algebraic topology deals with computation of the (Co)Homology and Homotopy groups of several spaces, while topology of the space is not actual for such computations. It seem be more Algebra than Topology, I did not see literature where are considered problems connected to the question. In my opinion Algebraic Topology studies above mentioned groups for Euclidean like spaces or their subspaces (manifolds), but in general topology we consider more general topological spaces (consisting of elements without algebraic operation of addition and multiplication) than Euclidean. I will be glad if my reason about AT will be rejected after examination of the books by Harer and Edelsbrunner and Fulton.

Http://arxiv.org/abs/math/0004133

this is an interesting article, a little speculative, but discuss some original and inspiring ideas.

The book "Algebraic topology of finite topological spaces and its applications" written by Jonathon Barmak (Springer-Verlag Berlin Heidelberg 2011) is a brilliant reference about your question.

Maybe this paper (and references therein) is of interest for you:

http://www.santafe.edu/media/workingpapers/01-02-011.pdf

Thanks Dr. Hellmuth,

at a glance it seem to be relative thing.

Dear Irakli: First I would like to comment on the relationship of Algebraic topology, to Topology itself. Point Set Topology, uses the ill defined concept of "point". See my article: Points, structures and levels of reality, in my RG page. Algebraic topology on the other tries to be point-free approach. By studying Homotopy, homology etc. groups, essentially reduces real topological problems into some algebraic ones. Instead of "point" we have simplicial comlexes. A good book for all these is:

Jakob Jonsson, Simplicial Complexes of Graphs, Spinger, 2008.

I should add to my previous answer, that trees give ordinary set theory, whereas general pointed graphs lead to Non-well-founded sets. I suppose that in this AST (Aczel Antifoundation Set Theory) one may develop topology. In this case there is a direct connection of topology and graph theory.

On the textbook of general topology: Munkres, James R.: "Topology". The title of the final chapter (Chap. 14) is "Applications to Graph Theory".

As another thing, I think there is a connection between graph theory and 'algebraic' topology via covering spaces with lifts (Galois correspondence).

Have also a look at:

http://en.wikipedia.org/wiki/Topological_graph_theory

My sincere thanks to all colleagues who suggested their answer(s).

The connection between graph theory and topology led to a sub field called topological graph theory. An important problem in this area concerns planar graphs.

I will suggest you those links:

reference request - Model Theory and Topology Connections ...

http://math.stackexchange.com/questions/262654/model-theory-...

Early Writings on Graph Theory: Topological Connections

http://www.math.nmsu.edu/hist_projects/Veblen-Barnett.pdf

A good overview of the connections between graph theory and general topology is given in

A. Vella, A Fundamentally Topological Perspective on Graph Theory, Ph.D. thesis, University of Waterloo, 2005:

http://etd.uwaterloo.ca/etd/avella2005.pdf

A good place to start is Section 1.1 (From Graph Theory to Topology). The story starts with one of the most important building blocks in graph theory, namely, connectedness.

Notice that connectedness is also at the threshold of Klette and Rosenberg, Digital Geometry.

Lot of Thanks Dear Prof. Peters and Gjolbanaj

Dear Qefsere,

The links you have given are excellent.   Another place to consider is convex sets in  topological vector spaces.   So, for example, a convex set A can either be an open or closed set and we can consider either the Cech or Kuratowski closure of A.  Graphically, a convex set can either have a polygonal shape or a nonlinear shaped derived from a B-spline curve on some of the vertices of adjacent polygons (e.g., a B-spline curve on the vertices in a Voronoi diagram).   

http://demonstrations.wolfram.com/BSplineCurvesFromVoronoiPolygonVertices/

The hats in the attached tessellation of a Kufi module are examples of polygons that are convex sets.  

Perhaps you know of other examples of other nonlinear convex sets in topological vector spaces.

I think additionally that, one of the interesting answers could be find in the paper by Claudia Malvenuto-  The Hopf algebra of finite topologies and T-partitions (see her RG page)

Article The Hopf algebra of finite topologies and T-partitions

In the paper below, the authors extend several results on Riemannian manifolds to graphs.  

Article Conformally Covariant Operators and Conformal Invariants on ...

Thank you very much dear Jonathan, the paper you recommend looks at glance interesting.