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 first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices. Alternatively, it is the sum of the degree sums d(i) + d(j) over all edges ij of the graph.
In the Maple program, T(n,k) gives the number of vertices of degree k in the Lucas cube Lambda(n).
LINKS
S. Klavžar, M. Mollard and M. Petkovšek, The degree sequence of Fibonacci and Lucas cubes, Discrete Mathematics, Vol. 311, No. 14 (2011), 1310-1322.
FORMULA
Conjectures from Colin Barker, Mar 28 2019: (Start)
G.f.: 2*x^2*(3 - 3*x + 2*x^2 + 3*x^4 - 2*x^5 - x^6) / (1 - x - x^2)^3.
a(n) = 3*a(n-1) - 5*a(n-3) + 3*a(n-5) + a(n-6) for n>8.
(End)
EXAMPLE
a(2) = 6 because the Lucas cube Lambda(2) is the path-tree P_3 having 2 vertices of degree 1 and 1 vertex of degree 2; consequently the first Zagreb index is 1^2 + 1^2 + 2^2 = 6 (or (1 + 2) + (2 + 1) = 6).
MAPLE
L:=(1+(1-y)*x + x^2*y^2 + (1-y)*x^3*y-(1-y)^2*x^4*y)/((1-x*y)*(1-x^2*y)-x^3*y): M:=expand(series(L, x=0, 40)): T:=(n, k)->coeff(coeff(M, x, n), y, k): Z1:=n->add(T(n, k)*k^2, k=0..n): seq(Z1(n), n=1..35);
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Mar 28 2019
STATUS
approved