OFFSET
3,1
COMMENTS
The Zagreb index of a graph is the sum of the squares of the degrees over all vertices 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.
I. Gutman and K. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), 83-92.
A. Hou, S. Li, L. Song, and B. Wei, Sharp bounds for Zagreb indices of maximal outerplanar graphs, J Comb Optim 22 (2011), 252-269.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = 3*(n-1)^2 + 9*(n-3).
a(n) = 6*A046691(n-2) for n>2.
a(n) = 6*A060577(n-1) for n>3.
G.f.: 6*x^3*(2 - x^2)/(1 - x)^3. - Stefano Spezia, Apr 12 2024
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 5. - Chai Wah Wu, Apr 16 2024
Sum_{n>=3} 1/a(n) = 19/72 + Pi*tan(Pi*sqrt(33)/2)*sqrt(33)/99 = 0.1865497.... - R. J. Mathar, Apr 22 2024
EXAMPLE
The graph K_3 has 3 degree 2 vertices, so a(3) = 3*4 = 12.
MATHEMATICA
Array[3*(#^2 + # - 8) &, 50, 3] (* Paolo Xausa, Jun 09 2024 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Allan Bickle, Apr 11 2024
STATUS
approved