A198341 The overall hyper-Wiener index of the rooted tree having Matula-Goebel number n.

0, 1, 7, 7, 28, 28, 30, 30, 84, 84, 84, 94, 94, 94, 210, 104, 94, 247, 104, 243, 243, 210, 247, 283, 462, 247, 579, 278, 243, 565, 210, 320, 462, 243, 547, 681, 283, 283, 565, 667, 247, 661, 278, 547, 1216, 579, 565, 793, 644, 1174, 547, 661, 320, 1506, 924

Offset: 1

Author

Emeric Deutsch, Dec 04 2011

Keywords

• nonn

Comments

The overall hyper-Wiener index of any connected graph G is defined as the sum of the hyper-Wiener indices of all the subgraphs of G. 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.

The Maple program yields a(n) by using the command OHW(n) for n<=3000 (adjustable).

Examples

			a(4)=7 because the rooted tree with Matula-Goebel number 4 is V; each of the 3 one-vertex subtrees has hyper-Wiener index 0, each of the 2 one-edge subtrees has hyper-Wiener index 1, and the tree V itself has hyper-Wiener index 5; 0+0+0+1+1+5=7.
		

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.
• D. Bonchev, The overall Wiener index - a new tool for characterization of molecular topology, J. Chem. Inf. Comput. Sci., 2001, 41, 582-592.
• X. H. Li and J. J. Lin, The overall hyper-Wiener index, J. Math. Chemistry, 33, 2003, 81-89.

Links

• E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A196060, A198339.

Programs

Maple

m2union := proc (x, y) sort([op(x), op(y)]) end proc: with(numtheory); MRST := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then [1] elif bigomega(n) = 1 then [1, seq(ithprime(mrst[pi(n)][i]), i = 1 .. nops(mrst[pi(n)]))] else [seq(seq(mrst[r(n)][i]*mrst[s(n)][j], i = 1 .. nops(mrst[r(n)])), j = 1 .. nops(mrst[s(n)]))] end if end proc: MNRST := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then [] elif bigomega(n) = 1 then m2union(mrst[pi(n)], mnrst[pi(n)]) else m2union(mnrst[r(n)], mnrst[s(n)]) end if end proc: MST := proc (n) m2union(mrst[n], mnrst[n]) end proc: for n to 3000 do mrst[n] := MRST(n): mnrst[n] := MNRST(n): mst[n] := MST(n) end do: 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: HW := proc (n) options operator, arrow: subs(x = 1, diff(W(n), x)+(1/2)*(diff(W(n), `$`(x, 2)))) end proc: OHW := proc (n) options operator, arrow: add(HW(MST(n)[j]), j = 1 .. nops(MST(n))) end proc: seq(OHW(n), n = 1 .. 60);

Formula

A198339(n) gives the sequence of the Matula-Goebel numbers of all the subtrees of the rooted tree with Matula-Goebel number n. A196060(k) is the hyper-Wiener index of the rooted tree with Matula-Goebel number k.