A212621 The overall first Zagreb index of the rooted tree having Matula-Goebel number n.

0, 2, 10, 10, 28, 28, 36, 36, 60, 60, 60, 80, 80, 80, 110, 112, 80, 158, 112, 146, 146, 110, 158, 222, 182, 158, 294, 196, 146, 266, 110, 320, 182, 146, 238, 414, 222, 222, 266, 370, 158, 354, 196, 238, 472, 294, 266, 594, 312, 424, 238, 354, 320, 744, 280, 494, 370, 266, 146, 660, 414, 182, 624

Offset

1,2

Comments

The overall first Zagreb index of any simple connected graph G is defined as the sum of the first Zagreb indices of all the subgraphs of G. The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices.

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.

References

D. Bonchev and N. Trinajstic, Overall molecular descriptors. 3. Overall Zagreb indices, SAR and QSAR in Environmental Research, 12, 2001, 213-236.

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 Y-N. 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.

Links

Table of n, a(n) for n=1..63.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.

Index entries for sequences related to Matula-Goebel numbers

Formula

A198339(n) gives the sequence of the Matula-Goebel numbers of all the subtrees of the rooted tree with Matula-Goebel number n. A196053(k) is the first Zagreb index of the rooted tree with Matula-Goebel number k.

Example

a(3)=10 because the rooted tree with Matula-Goebel number 3 is the path tree with 3 vertices R - A - B; the subtrees are R, A, B, RA, AB, and RAB with first Zagreb indices 0, 0, 0, 2, 2, and 6, respectively.

Maple

with(numtheory); Z1 := 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 0 elif bigomega(n) = 1 then Z1(pi(n))+2+2*bigomega(pi(n)) else Z1(r(n))+Z1(s(n))-bigomega(r(n))^2-bigomega(s(n))^2+bigomega(n)^2 end if end proc; 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; OZ1 := proc (n) options operator, arrow; add(Z1(MST(n)[j]), j = 1 .. nops(MST(n))) end proc; seq(OZ1(n), n = 1 .. 120); # MRST considers the subtrees that contain the root; MNRST considers the subtrees that do not contain the root; MST considers all subtrees.

Crossrefs

Cf. A198339, A196053, A212618, A212619, A212620, A212622, A212623, A212624, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.

Sequence in context: A071808 A355486 A168381 * A156780 A206486 A067046

Adjacent sequences: A212618 A212619 A212620 * A212622 A212623 A212624

Keyword

nonn

Author

Emeric Deutsch, Jun 01 2012

Status

approved