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A372025
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Maximum second Zagreb index of maximal 3-degenerate graphs with n vertices.
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4
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12, 54, 120, 210, 324, 462, 624, 810, 1020, 1254, 1512, 1794, 2100, 2430, 2784, 3162, 3564, 3990, 4440, 4914, 5412, 5934, 6480, 7050, 7644, 8262, 8904, 9570, 10260, 10974, 11712, 12474, 13260, 14070, 14904, 15762, 16644, 17550, 18480, 19434, 20412, 21414, 22440, 23490, 24564, 25662, 26784, 27930
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OFFSET
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3,1
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = 3*(n-1)^2 + 9*(n-3)*(n-1).
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
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EXAMPLE
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The graph K_3 has 3 degree 2 vertices, so a(3) = 3*4 = 12.
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CROSSREFS
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Cf. A372027 (second Zagreb index of MOPs).
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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