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 outerplanar graph has all vertices on the exterior region, and all other regions triangles. The extremal graphs are fans, except when n=6. Then the extremal graph is the triangular grid with degrees 4,4,4,2,2,2.
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.
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^2 + n - 19 when n is not 3 or 6.
From Chai Wah Wu, Apr 16 2024: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 9.
G.f.: x^3*(x^6 - 3*x^5 + 3*x^4 + 2*x^2 + 3*x - 12)/(x - 1)^3. (End)
EXAMPLE
The graph K_3 has 3 degree 2 vertices, so a(3) = 3*4 = 12.
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {12, 33, 61, 96, 135, 181, 233}, 50] (* Paolo Xausa, Jan 22 2025 *)
PROG
(PARI) a(n)=if(n>6, 3*n^2+n-19, [12, 33, 61, 96][n-2]) \\ Charles R Greathouse IV, May 26 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Allan Bickle, Apr 16 2024
STATUS
approved