

A212621


The overall first Zagreb index of the rooted tree having MatulaGoebel number n.


10



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
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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 MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel 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, 213236.
F. Goebel, On a 11 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YN. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
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 MatulaGoebel numbers


FORMULA

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


EXAMPLE

a(3)=10 because the rooted tree with MatulaGoebel 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))^2bigomega(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: A156556 A071808 A168381 * A156780 A206486 A067046
Adjacent sequences: A212618 A212619 A212620 * A212622 A212623 A212624


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Jun 01 2012


STATUS

approved



