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%I #2 Apr 19 2016 01:16:09
%S 1,1,2,4,10
%N Number of fixed points of N_{2,2}(G_n).
%C G_n are connected graphs of n nodes.
%C N_{m,n} is a mapping form k nodes graph to k nodes graph. N is for "Near" [Definition] For all pairs of distinct vertices x,y in G if n paths of length m exist between x and y then add an edge xy. The graph H which is made from G is represented as N_{m,n}(G).
%e Example: N_{1,1}(G)=G. Other definition of N_{2,3}: G={V,E_g}, H=N_{2,3}(G), H={V,E_h}. All x,y (x,y E V and -x=y and (Exist p,q,r -p=q and -p=r and -q=r and xp,xq,xr,py,qy,ry E E_g)) - E_h = E_g U {xy} where "E" means "element" and "-" means "not"
%e Fixed points of N_{2,2}: n = number of nodes. We count only connected graphs.
%e n=1
%e ....o
%e n=2
%e ....o_o
%e n=3
%e ....o_o_o....o_o
%e .............|/
%e .............o
%e n=4
%e ....o_o_o_o....o_o_o....o_o_o....o_o
%e .................|......|/.......|x|
%e .................o......o........o_o
%e n=5
%e ....o_o_o_o_o....o_o_o_o....o_o_o_o....o_o_o......o_o_o_o
%e ...................|........|/.........|...|........|/...
%e ...................o........o..........o___o........o....
%e .....................................................
%e .........o_o_o.....o_o_o.....o_o_o......o_o_o....o_o_o
%e ........../|.......|/|.......|/.........|x|......|x-x|
%e .........o.o.......o.o.......o_o........o_o......o___o
%e ..................................................K_5
%e .........o_o....o_o
%e .........|.|....|/|
%e .........o_o....o_o
%e These graphs don't have the following subgraphs:
%e o_o ... o_o
%e | | ... |/|
%e o_o ... o_o
%K nonn,uned
%O 1,3
%A _Yasutoshi Kohmoto_, Feb 18 2008
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