OFFSET
1,2
COMMENTS
A tree is said to be central if its center consists of one point (see the Harary reference, p. 35).
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).
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 not 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.
EXAMPLE
7 is in the sequence because the rooted tree with Matula-Goebel number 7 is Y which is central. 5 is not in the sequence because the corresponding rooted tree is the path-tree on 4 vertices, a bicentral tree.
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: C := {}: for n to 130 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) = false then C := `union`(C, {n}) else end if end do: C;
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Nov 24 2011
STATUS
approved