%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