@Hesam Moslemzadeh: we use it in analysing structures and forming their incident matrixes...

This dovetails with my principal uses of graph theory.    In my case, graph theory is handy in representing connectedness, nerve, and boundedness structures in either Voronoi or Delaunay meshes.

we use it in analysing structures and forming their incident matrixes and using it's null spaces for mechanisms and state of indeterminacies.tensegrity structures are one of the most used areas of it's applications.

@Hesam Moslemzadeh: we use it in analysing structures and forming their incident matrixes...

This dovetails with my principal uses of graph theory.    In my case, graph theory is handy in representing connectedness, nerve, and boundedness structures in either Voronoi or Delaunay meshes.

There are a lot of application of graph theory in automata theory, control systems theory, optimal control, distributed control....! I may give you a lot of examples about dear @Sudev! Mason's rules in algebra of block diagram of complex systems is well known long time ago.

Also, it is used in some compiler design and operating system applications in computer sciences

I agree with Ljubomir and Qasim.

In flow networks spanning trees are associated with a set of linearly independent columns in a network incidence matrix, describing for each arc/link which are the starting and ending nodes, for example. A cycle in the network is associated with a set of linearly dependent associated such column vectors. Feasible changes of the vector of flows in the network (with respect to flow balance in the nodes and the amount of flows emanating from the sources and terminating at the sinks can then be described by movements in cycles (whose columns are linearly dependent, and describe algebraically a movement in the null space of the graph-defining matrix). These relations also to some degree explain the effectiveness of linear programming based algorithms for network flow problems.

Many such relations can be found, and often well explained, in basic text books on network flows, such as the excellent one by Ahuja, Magnanti and Orlin.

An example of Google using graph theory

http://math.uchicago.edu/~shmuel/Network-course-readings/fea-chung.pdf

This is very good article dear @Krishnan, bringing the application of graph theory in fast searching algorithms.

Graph theory is an important tool of mathematics in different applicable fields.  Connectedness problems (where topology arises), traceability, networking and flowcharts, computer science, neuron structures , creating ordered relations between objects and studying their properties in structures of science or reality. 

Dear Professor Sudev Naduvath,

Graph theory has wide range of applications; and it has been successfully applied in many problems of Operations Management .

Best Regards.

An application in communication networks- switch count and congestion using 2D array, binary tree

http://www.ijaiem.org/volume1Issue2/IJAIEM-2012-10-11-017.pdf

In 2D or in 3D plane for find a fix point or fix area graph Theory is the best way.

This fix point or fix area may be define any thing.

There is a good content for direct use - Tom Davis (2005), Graph Theory Problems and Solutions

http://www.geometer.org/mathcircles/graphprobs.pdf

Please check the following links for some applications of Graph Theory. Hope that these articles will be useful to you. 

http://manufacturingscience.asmedigitalcollection.asme.org/article.aspx?articleid=1439722

http://pubs.acs.org/doi/pdf/10.1021/ed065p574

http://www.dbpia.co.kr/Journal/ArticleDetail/827189

http://link.springer.com/article/10.1007/BF02291704#page-1

http://www.sciencedirect.com/science/article/pii/030146229080056D

http://www.sciencedirect.com/science/article/pii/S0301479700903736

http://www.sciencedirect.com/science/article/pii/S0022247X00972433

http://onlinelibrary.wiley.com/doi/10.1111/j.1752-4571.2008.00047.x/full

http://pubs.acs.org/doi/abs/10.1021/cr00070a005

Fine graph applications examples in different fields were brought by dear @Chitra. Thanks!

Dear All,

Thank you so much for your valuable remarks. I appreciate your inputs very much..

Sudev

Dear Sudev: I studied the theory of graphs, but restricted mostly to planar graphs, trees and forests and I applied the theory for generating feasible solutions for the layout in plant of sanitary sewerage systems and of drainage pluvial systems. It is a passionate area. I developed several algorithms that combined graph theory with several optimization techniques. Two of my more important research papers (that you can find in my RG page) include the description of such algorithms.