A198331 Matula-Goebel numbers of bicentral rooted trees.

2, 5, 6, 12, 13, 14, 15, 17, 22, 24, 28, 30, 31, 35, 37, 38, 39, 43, 44, 45, 47, 48, 51, 55, 56, 58, 60, 67, 69, 70, 76, 78, 79, 82, 88, 89, 90, 91, 93, 94, 95, 96, 102, 105, 106, 107, 109, 110, 111, 112, 113, 116, 117, 118, 119, 120, 129, 135, 138, 140, 142

Offset: 1

Author

Emeric Deutsch, Nov 24 2011

Keywords

• nonn

Comments

A tree is said to be bicentral if its center consists of two points (see the Harary reference, p. 35).

Examples

			5 is in the sequence because the rooted tree with Matula-Goebel number 5 is the path-tree on 4 vertices which is bicentral.
		

References

• F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
• I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
• I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
• D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
• A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
• J. M. Aldous and R. J. Wilson, Graphs and Applications, Springer, 2000 (p. 179).

Links

• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A198329, A198330.

Programs

Maple

with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow; n/r(n) end proc: b := proc (n) if n = 1 then 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(b(pi(n))) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then b(pi(n)) else b(r(n))*b(s(n)) end if end proc: CC := {}: for n to 180 do B := {n}: nn := n: for i while a(nn) > 1 do B := `union`(B, {a(nn)}): nn := a(nn) end do: if member(2, B) = true then CC := `union`(CC, {n}) else  end if end do: CC;

Formula

A198329(n) is the Matula-Goebel number of the rooted tree obtained by removing from the rooted tree with Matula-Goebel number n the vertices of degree one, together with their incident edges. If the repeated application of this pruning operation will lead to the Matula-Goebel number 2 (corresponding to the 1-edge tree), then the starting rooted tree is central. The Maple program is based on this.