A198340 The overall Wiener index of the rooted tree having Matula-Goebel number n.

0, 1, 6, 6, 21, 21, 24, 24, 56, 56, 56, 67, 67, 67, 126, 80, 67, 161, 80, 154, 154, 126, 161, 197, 252, 161, 354, 188, 154, 333, 126, 240, 252, 154, 311, 440, 197, 197, 333, 414, 161, 411, 188, 311, 683, 354, 333, 545, 384, 636, 311, 411, 240, 921, 462, 510

Offset: 1

Author

Emeric Deutsch, Dec 04 2011

Keywords

• nonn

Comments

The overall Wiener index of any connected graph G is defined as the sum of the Wiener indices of all the subgraphs of G. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.

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(4)=6 because the rooted tree with Matula-Goebel number 4 is V; each of the 3 one-vertex subtrees has Wiener index 0, each of the 2 one-edge subtrees has Wiener index 1, and the tree V itself has Wiener index 4; 0+0+0+1+1+4=6.
		

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.

Links

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

Crossrefs

Cf. A196051, 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 2000 do mrst[n] := MRST(n): mnrst[n] := MNRST(n): mst[n] := MST(n) end do: W := proc (n) local u, v, E, PL: u := proc (n) options operator, arrow: op(1, factorset(n)) end proc: v := proc (n) options operator, arrow: n/u(n) end proc: E := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n)) else E(u(n))+E(v(n)) end if end proc: PL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n))+PL(pi(n)) else PL(u(n))+PL(v(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then W(pi(n))+PL(pi(n))+1+E(pi(n)) else W(u(n))+W(v(n))+PL(u(n))*E(v(n))+PL(v(n))*E(u(n)) end if end proc: OW := proc (n) options operator, arrow: add(W(MST(n)[j]), j = 1 .. nops(MST(n))) end proc: seq(OW(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. A196051(k) is the Wiener number of the rooted tree with Matula-Goebel number k.