A196060 The hyper-Wiener index of the rooted tree with Matula-Goebel number n.

0, 1, 5, 5, 15, 15, 12, 12, 35, 35, 35, 28, 28, 28, 70, 22, 28, 54, 22, 58, 58, 70, 54, 44, 126, 54, 90, 47, 58, 99, 70, 35, 126, 58, 108, 76, 44, 44, 99, 84, 54, 83, 47, 108, 150, 90, 99, 63, 91, 165, 108, 83, 35, 118, 210, 69, 84, 99, 58, 131, 76, 126, 129, 51, 170, 170, 44, 91, 150, 143, 84, 101, 83, 76, 231

Offset: 1

Author

Emeric Deutsch, Sep 30 2011

Keywords

• nonn

Comments

The hyper-Wiener index of a connected graph is (1/2)*Sum [d(i,j)+d(i,j)^2], where d(i,j) is the distance between the vertices i and j and summation is over all unordered pairs of vertices (i,j).

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.

Examples

			a(7)=12 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y; the distances are 1,1,1,2,2,2; sum of distances = 9; sum of squared distances = 15; (9+15)/2=12.
a(2^m) = m(3m-1)/2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges and we have m distances 1 and m(m-1)/2 distances 2; sum of the  distances = m^2; sum of the squared distances = 2m^2 - m; hyper-Wiener index is (1/2)(3m^2 - m).
		

Links

• G. G. Cash, Relationship between the Hosoya polynomial and the hyper-Wiener index, Appl. Math. Letters, 15, 2002, 893-895.
• Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
• 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. J. Klein, I. Lukovits and I. Gutman, On the definition of the hyper-Wiener index for cycle-containing structures, J. Chem. Inf. Comput. Sci., 35, 1995, 50-52.
• D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A196059.

Programs

Maple

with(numtheory): W := 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(W(pi(n))+x*R(pi(n))+x)) else sort(expand(W(r(n))+W(s(n))+R(r(n))*R(s(n)))) end if end proc: a := proc (n) options operator, arrow: subs(x = 1, diff(W(n), x)+(1/2)*(diff(W(n), `$`(x, 2)))) end proc: seq(a(n), n = 1 .. 75);

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]]];
W[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, W[PrimePi[n]] + x*R[PrimePi[n]] + x, True, W[r[n]] + W[s[n]] + R[r[n]]*R[s[n]]];
a[n_] := (D[W[n], x] /. x -> 1) + (1/2)*(D[W[n], {x, 2}] /. x -> 1);
Table[a[n], {n, 1, 75}] (* Jean-François Alcover, Jun 19 2024, after Maple code *)

Formula

a(n) = W'(n,1) + (1/2)W"(n,1), where W(n,x) is the Wiener polynomial (also called Hosoya polynomial) of the rooted tree with Matula-Goebel index n. W(n)=W(n,x) is obtained recursively in A196059. The Maple program is based on the above.