OFFSET
1,3
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.
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.
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.
EXAMPLE
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).
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Sep 30 2011
STATUS
approved