Hello,
I need to know what is the benefict of using the graph theory for solving some mathematic problems, ( when we projected them in this space) and what's the best references of graph theory and its applications.
Ok, you would need to tell us more exactly. I am a bit puzzled by your questions. Graph theory is part of combinatorics. It is often studied in under-graduate modules in mathematics degree.
As you do not provide more information. I would recommend you to look at Cartesian Genetic programming. It can solve some mathematical problems by using evolution on a graph-based encoding scheme.
It is not fully clear what do you mean by "graph space". I understand it in a specific way giving the answer a possible quite wrong direction.
A continuous operator A mapping a Hilbert or Banach space X to another (or even the same) one Y is called closed if the set of all (x,A(x)) pairs is a closed subset of the product space X*Y, the graph space of the operator. (This concept make sense for infinite dimensional spaces only since operators in finite dimensional spaces are always closed.) A prominent example is the differential operator mapping a function f (say over the real line with variable t) to its derivative d/dt f. The advantage of closed operators is as follows. Given a sequence of points x_n in X and its images y_n=A(x_n) such that limits x_0 and y_0, resp., exists then A(x_0)=y_0 holds (this equality does not always hold for non-closed operators). For example, polynomials x_n=(1+t/n)^n are differentiable and converge to e^t. Since the derivatives d/dt ((1+t/n)^n) also tend do e^t, function e^t is differentiable and its derivative is again e^t. So far to the advantage of closed operators. The advantage of graph spaces is to help establishing the important limit relation for a given operator.
Unfortunately, my knowledge of related textbooks is outdated (the one in my bookshelf was printed in 1968).
BTW, theory of graph spaces (in the sense of this answer) is not graph theory.
In my opinion we can solve some mathematic problems by projecting this problem to the "graph space" like you described it and extract some features of the solution through the search for the shortest path or cycles.
Projection to the graph space is really useful like, for instance, for the internet ranking problem (pagerank) or ressources allocations (to search possible mutual blocking by searching cycles).
I'm very interested by neuroscience and look forward to how the graph structure of the cortex (and hippocampus) can solve our daily problems.
Dear teachers,
I appreciate all of your answers.Your answer is rich and valuable.
Thank you.
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