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%I #13 Apr 22 2024 07:09:07
%S 12,54,120,210,324,462,624,810,1020,1254,1512,1794,2100,2430,2784,
%T 3162,3564,3990,4440,4914,5412,5934,6480,7050,7644,8262,8904,9570,
%U 10260,10974,11712,12474,13260,14070,14904,15762,16644,17550,18480,19434,20412,21414,22440,23490,24564,25662,26784,27930
%N Maximum second Zagreb index of maximal 3-degenerate graphs with n vertices.
%C The second Zagreb index of a graph is the sum of the products of the degrees over all edges of the graph.
%C 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.
%H Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.
%H Allan Bickle, <a href="https://ajc.maths.uq.edu.au/pdf/89/ajc_v89_p167.pdf">Zagreb Indices of Maximal k-degenerate Graphs</a>, Australas. J. Combin. 89 1 (2024) 167-178.
%H J. Estes and B. Wei, <a href="https://doi.org/10.1007/s10878-012-9515-6">Sharp bounds of the Zagreb indices of k-trees</a>, J Comb Optim 27 (2014), 271-291.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 3*(n-1)^2 + 9*(n-3)*(n-1).
%F From _Chai Wah Wu_, Apr 16 2024: (Start)
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 5.
%F G.f.: x^3*(6*x^2 - 18*x - 12)/(x - 1)^3. (End)
%F 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
%e The graph K_3 has 3 degree 2 vertices, so a(3) = 3*4 = 12.
%Y Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).
%Y Cf. A051624, A372025, A372026 (second Zagreb indices of maximal k-degenerate graphs).
%Y Cf. A372027 (second Zagreb index of MOPs).
%K nonn,easy
%O 3,1
%A _Allan Bickle_, Apr 16 2024
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