A186310 Total Wiener index of star-like trees with n edges.

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

Offset: 1

Author

Washington Bomfim, Feb 17 2011

Keywords

• nonn

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

Examples

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

Links

• Washington Bomfim, Table of n, a(n) for n = 1..140
• Washington Bomfim, Example
• Arnold Knopfmacher, Robert F. Tichy, Stephan Wagner, and Volker Ziegler, Graphs, Partitions and Fibonacci Numbers
• Stephan Wagner, Graph-theoretical enumeration and digital expansions: an analytic approach, Dissertation, Fakult. f. Tech. Math. u. Tech. Physik, Tech. Univ. Graz, Austria, Feb. 2006.

Crossrefs

Cf. A001620, A002117, A058984, A122681.