Theory NO. 5 : Fibbonacci strikes again!
18Feb09
Today, I noticed an amazing and worth memorizing thing. I turns out that fibbonacci series shows up also in graph theory in one of the simplest graphs:
Where’s Fibbonacci hiden here? Well, the graph can be represented as a folowing matrix:
![M={\left[\begin{array}{ccc} 1 & 1 \\ 1 & 0 \\ \end{array}\right]}](http://s0.wp.com/latex.php?latex=M%3D%7B%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D+1+%26+1+%5C%5C+1+%26+0+%5C%5C+%5Cend%7Barray%7D%5Cright%5D%7D+&bg=ffffff&fg=333333&s=0 "M={\left[\begin{array}{ccc} 1 & 1 \\ 1 & 0 \\ \end{array}\right]}")
By multiplying the matrix by itself and using induction we get:
![M^{n} ={\left[\begin{array}{ccc} FIB(n) & FIB(n-1) \\ FIB(n-1) & FIB(n-2) \\ \end{array}\right]}](http://s0.wp.com/latex.php?latex=M%5E%7Bn%7D+%3D%7B%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D+FIB%28n%29+%26+FIB%28n-1%29+%5C%5C+FIB%28n-1%29+%26+FIB%28n-2%29+%5C%5C+%5Cend%7Barray%7D%5Cright%5D%7D+&bg=ffffff&fg=333333&s=0 "M^{n} ={\left[\begin{array}{ccc} FIB(n) & FIB(n-1) \\ FIB(n-1) & FIB(n-2) \\ \end{array}\right]}")
Filed under: Mathematics, Science, Theory | 6 Comments
I haven’t got how does the matrix represent the Graph??
The matrix is the representation of the graph in the simplest posible way:
then it means that between verticles 
and 
exists a path (edge).
has a loop, then it has 1 in 
on axis.
The index of any column or row represents a verticle, and if
It’s also easy to conclude that if a vericle
When you’ll multiply matrix
by itself 2 times, then numbers in the matrix represent the number of paths which length is 2 between choosen 2 verticles.
When you’ll multiply matrix
by itself 3 times, then numbers in the matrix represent the number of paths which length is 3 between choosen 2 verticles.
ooh Graph .. I got it now
I was thinking in it as a Circle and line
Thanks
Maybe you’ll analyse some topology and share your insights?
Just passing by.Btw, your website have great content!
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