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A186310 Total Wiener index of star-like trees with n edges. 0
1, 4, 19, 54, 145, 350, 726, 1462, 2710, 4846, 8166, 13730, 21741, 34350, 52488, 79518, 117141, 172224, 246993, 353464, 496721, 694952, 958374, 1318174, 1789077, 2420888, 3243045, 4329098, 5728140, 7557906, 9893327, 12913820, 16746136 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

In the reference, p. 18, theorem 2.14, there is the following formula of the average Wiener index av(n) of a star-like tree with n edges:

av(n) = 2*n^2 - (6^(1/2)*n^(3/2))/(2*Pi)*(log(n) + 2*cEuler - log(Pi^2/6) + 24*zeta(3)/(Pi^2)),

so an approximate value of a(n) is given by av(n)*A058984(n). The following table was determined approximating zeta(3) by 1.2020569, and Euler's constant by 0.5772156649.

    n  av(n)*A058984(n) (I)     a(n) (II)     I/II

    5             136.9              145   0.94414

   13           21443.1            21741   0.98630

   20          352132.8           353464   0.99623

   28         4329081.3          4329098   0.999996

   29         5729910.2          5728140   1.00031

   30         7560843.8          7557906   1.00039

   33        16760543.2         16746136   1.00086

   50       810144542.2        808929430   1.00150

   60      5614575632.9       5606027232   1.00152

   80    167110984160.2     166870656888   1.00144

  100   3203299185861.4    3199052703248   1.00133

  120  45208751880788.8   45153537110230   1.00122

  130 155331813239050.0  155149438632558   1.00117

  140 507674790104504.3  507101038817616   1.00113

For n<=28 the approximation underestimates the actual value of the total Wiener index of star-like trees. For 29 <= n <= 140 it overestimates this total; however as n grows, the rate I/II converges to 1. - Washington Bomfim, Feb 17 2011

REFERENCES

S. Wagner, Graph-theoretical enumeration and digital expansions: an analytic approach, Dissertation, Fakult. f. Tech. Math. u. Tech. Physik, Tech. Univ. Graz, Austria, Feb. 2006.

LINKS

Table of n, a(n) for n=1..33.

Washington Bomfim, Example

Washington Bomfim, Table of n, a(n) for n=1..140

Arnold Knopfmacher, Robert F. Tichy, Stephan Wagner, and Volker Ziegler, Graphs, Partitions and Fibonacci Numbers

EXAMPLE

The first Bomfim link shows a way to find a(7).

CROSSREFS

Cf. A001620, A002117, A058984, A122681.

Sequence in context: A159833 A166808 A253008 * A122684 A122681 A215862

Adjacent sequences:  A186307 A186308 A186309 * A186311 A186312 A186313

KEYWORD

nonn

AUTHOR

Washington Bomfim, Feb 17 2011

STATUS

approved

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Last modified July 22 20:51 EDT 2019. Contains 325226 sequences. (Running on oeis4.)