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A212632 The domination number of the rooted tree with Matula-Goebel number n. 15
1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 2, 3, 1, 2, 2, 2, 3, 2, 3, 3, 3, 2, 2, 3, 2, 1, 3, 2, 2, 3, 2, 2, 3, 2, 3, 3, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 1, 4, 3, 2, 2, 3, 2, 3, 3, 3, 3, 1, 3, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 2, 3, 4, 2, 2, 4, 3, 3, 3, 3, 3, 3, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

The domination number of a simple graph G is the minimum cardinality of a dominating subset of G.

The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.

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 Y-N. 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.

É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.

LINKS

Table of n, a(n) for n=1..88.

S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, arXiv:0905.2251.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.

Index entries for sequences related to Matula-Goebel numbers

FORMULA

In A212630 one gives the domination polynomial P(n)=P(n,x) of the rooted tree with Matula-Goebel number n. We have a(n) = least exponent in P(n,x).

EXAMPLE

a(5)=2 because the rooted tree with Matula-Goebel number 5 is the path tree R - A - B - C; {A,B} is a dominating subset and there is no dominating subset of smaller cardinality.

MAPLE

with(numtheory): P := proc (n) local r, s, A, B, C:

r := n -> op(1, factorset(n)): s := n-> n/r(n):

A := proc (n) if n = 1 then x elif bigomega(n) = 1 then x*(A(pi(n))+B(pi(n))+C(pi(n))) else A(r(n))*A(s(n))/x end if end proc:

B := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then A(pi(n)) else sort(expand(B(r(n))*B(s(n))+B(r(n))*C(s(n))+B(s(n))*C(r(n)))) end if end proc:

C := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then B(pi(n)) else expand(C(r(n))*C(s(n))) end if end proc:

sort(expand(A(n)+B(n))) end proc:

A212632 := n->degree(P(n))-degree(numer(subs(x = 1/x, P(n)))): seq(A212632(n), n = 1 .. 120);

MATHEMATICA

A[n_] := Which[n == 1, x, PrimeOmega[n] == 1, x*(A[PrimePi[n]] + B[PrimePi[n]] + c[PrimePi[n]]), True, A[r[n]]*A[s[n]]/x];

B[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, A[PrimePi[n]], True, Expand[B[r[n]]*B[s[n]] + B[r[n]]*c[s[n]] + B[s[n]]*c[r[n]]]];

c[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, B[PrimePi[n]], True, Expand[c[r[n]]*c[s[n]]]];

r[n_] :=  FactorInteger[n][[1, 1]];

s[n_] := n/r[n];

P[n_] := Expand[A[n] + B[n]];

a[n_] := Exponent[P[n], x] - Exponent[Numerator[P[n] /. x -> 1/x // Together], x];

Array[a, 100] (* Jean-François Alcover, Nov 14 2017, after Emeric Deutsch *)

CROSSREFS

Cf. A212618 - A212631.

Sequence in context: A254687 A182590 A047846 * A025885 A198337 A206483

Adjacent sequences:  A212629 A212630 A212631 * A212633 A212634 A212635

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jun 11 2012

STATUS

approved

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Last modified January 22 23:00 EST 2019. Contains 319365 sequences. (Running on oeis4.)