A196066 The reverse Wiener index of the rooted tree with Matula-Goebel number n.

0, 0, 2, 2, 8, 8, 3, 3, 20, 20, 20, 12, 12, 12, 40, 4, 12, 29, 4, 28, 28, 40, 29, 17, 70, 29, 36, 16, 28, 55, 40, 5, 70, 28, 53, 40, 17, 17, 55, 38, 29, 38, 16, 53, 68, 36, 55, 23, 36, 93, 53, 38, 5, 48, 112, 21, 38, 55, 28, 73, 40, 70, 45, 6, 92, 92, 17, 36, 68, 70, 38, 53, 38, 40, 114, 21, 89, 72, 53, 50

Offset: 1

Author

Emeric Deutsch, Oct 01 2011

Keywords

• nonn

Comments

The reverse Wiener index of a connected graph is (1/2)N(N-1)D - W, where N, D, and W are, respectively, the number of vertices, the diameter, and the Wiener index of the graph.

The Matula-Goebel number of a rooted tree is 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.

Examples

			a(7)=3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y with N=4, d=2, W=9 (distances are 1,1,1,2,2,2); (1/2)*4*3*2-9 = 3.
		

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, D. Mills, O. Ivanciuc, and S. C. Basak, Reverse Wiener indices, Croatica Chemica Acta, 73 (4), 2000, 923-941.

Links

• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A196059.

Programs

Maple

with(numtheory): Wp := proc (n) local r, s, R: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: R := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*R(pi(n))+x)) else sort(expand(R(r(n))+R(s(n)))) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(Wp(pi(n))+x*R(pi(n))+x)) else sort(expand(Wp(r(n))+Wp(s(n))+R(r(n))*R(s(n)))) end if end proc: N := proc (n) options operator, arrow: 1+coeff(Wp(n), x) end proc: d := proc (n) options operator, arrow: degree(Wp(n)) end proc: W := proc (n) options operator, arrow: subs(x = 1, diff(Wp(n), x)) end proc: a := proc (n) options operator, arrow: (1/2)*N(n)*(N(n)-1)*d(n)-W(n) end proc: 0, seq(a(n), n = 2 .. 80);

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
R[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, x*R[PrimePi[n]] + x, True,  R[r[n]] + R[s[n]]];
Wp[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, Wp[PrimePi[n]] + x*R[PrimePi[n]] + x, True, Wp[r[n]] + Wp[s[n]] + R[r[n]]*R[s[n]]];
V[n_] := 1 + Coefficient[Wp[n], x];
d[n_] := Exponent[Wp[n], x];
W[n_] := D[Wp[n], x] /. x -> 1;
a[n_] := If[n == 1, 0, (1/2)*V[n]*(V[n] - 1)*d[n] - W[n]];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 22 2024, after Maple code *)

Formula

a(n)=(1/2)N(n)*(N(n)-1)*d(n) - W(n), where N, d, and W are, respectively, the number of vertices, the diameter, and the Wiener index of the rooted tree with Matula-Goebel number n (all these data are contained in the Wiener polynomial; see A196059). The Maple program is based on the above.