OFFSET
3,1
COMMENTS
The second Zagreb index of a graph is the sum of the products of the degrees over all edges of the graph.
A maximal 3-degenerate graph can be constructed from a 3-clique by iteratively adding a new 3-leaf (vertex of degree 3) adjacent to three existing vertices. The extremal graphs are 3-stars, so the bound also applies to 3-trees.
LINKS
Paolo Xausa, Table of n, a(n) for n = 3..10000
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Allan Bickle, Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin. 89 1 (2024) 167-178.
J. Estes and B. Wei, Sharp bounds of the Zagreb indices of k-trees, J Comb Optim 27 (2014), 271-291.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 3*(n-1)^2 + 9*(n-3)*(n-1).
From Chai Wah Wu, Apr 16 2024: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 5.
G.f.: x^3*(6*x^2 - 18*x - 12)/(x - 1)^3. (End)
a(n) = 6*A014107(n-1). Sum_{n>=3} 1/a(n) = (1/2+log(2))/9 = 0.1325719... - R. J. Mathar, Apr 22 2024
EXAMPLE
The graph K_3 has 3 degree 2 vertices, so a(3) = 3*4 = 12.
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {12, 54, 120}, 50] (* Paolo Xausa, Jan 22 2025 *)
CROSSREFS
Cf. A372027 (second Zagreb index of MOPs).
KEYWORD
nonn,easy
AUTHOR
Allan Bickle, Apr 16 2024
STATUS
approved