

A306823


a(n) is the second multiplicative Zagreb index of the Lucas cube Lambda(n).


1




OFFSET

1,2


COMMENTS

The Lucas cube Lambda(n) can be defined as the graph whose vertices are the binary strings of length n without either two consecutive 1's or a 1 in the first and in the last position, and in which two vertices are adjacent when their Hamming distance is exactly 1.
The second multiplicative Zagreb index of a simple connected graph is product(deg(x)^(deg(x)) over all the vertices of the graph (see, for example, the I. Gutman reference, p. 16).
In the Maple program G = Sum_{n>=0} P[n]z^n is the generating function of the Lucas cubes according to size (coded by z) and vertex degrees (coded by t). See the Klavzar  Mollard  Petkovsek reference: l(x,y) on p. 1321 with different variables.


LINKS



EXAMPLE

a(2)=4 because the Lucas cube Lambda(2) is the pathtree P_3 having 2 vertices of degree 1 and 1 vertex of degree 2; consequently, a(2) = 1^1*1^1*2^2 = 4.
a(4)=1048576 because the Lucas cube Lambda(4) is a bouquet of tw 4cycles, having 6 vertices of degree 2 and 1 vertex of degree 4; consequently, a(4) = (2^2)^6*4^4 = 2^12*v^4 = 1048576.


MAPLE

G:=(1+(1t)*z+t^2*z^2+t*(1t)*z^3t*(1t)^2*z^4)/((1t*z)*(1t*z^2)t*z^3): Gser:=simplify(series(G, z=0, 50)): for n from 0 to 45 do P[n]:=sort(coeff(Gser, z, n)) od: seq(product(j^(j*coeff(P[n], t, j)), j=1..n), n=1..7);


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



