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A212622 The overall second Zagreb index of the rooted tree having Matula-Goebel number n. 10
0, 1, 6, 6, 19, 19, 24, 24, 44, 44, 44, 59, 59, 59, 85, 80, 59, 125, 80, 114, 114, 85, 125, 173, 146, 125, 246, 156, 114, 219, 85, 240, 146, 114, 193, 344, 173, 173, 219, 302, 125, 297, 156, 193, 407, 246, 219, 481, 256, 360, 193, 297, 240, 651, 231, 414, 302, 219, 114, 567, 344, 146, 548, 672, 345, 345, 173, 256, 407, 482, 302, 914, 297 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The overall second Zagreb index of any simple connected graph G is defined as the sum of the second Zagreb indices of all the subgraphs of G. The second Zagreb index of a simple connected graph G is the sum of the degree products d(i)d(j) over all edges ij of g.

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..73.

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. A196054(k) is the second Zagreb index of the rooted tree with Matula-Goebel number k.

EXAMPLE

a(3)=6 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 second Zagreb indices 0, 0, 0, 1, 1, and 4, respectively.

MAPLE

with(numtheory): Z2 := proc (n) local r, s, a: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: a := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+bigomega(pi(n)) else a(r(n))+a(s(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then Z2(pi(n))+a(pi(n))+bigomega(pi(n))+1 else Z2(r(n))+Z2(s(n))+a(r(n))*bigomega(s(n))+a(s(n))*bigomega(r(n)) 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: OZ2 := proc (n) options operator, arrow: add(Z2(MST(n)[j]), j = 1 .. nops(MST(n))) end proc: seq(OZ2(n), n = 1 .. 120);

CROSSREFS

Cf. A098339, A196054, A212618, A212619, A212620, A212621, A212623, A212624, A212625, A212626, A212627, A212628, A212629, A212630, A212631, A212632.

Sequence in context: A294669 A224711 A073096 * A255468 A246037 A045896

Adjacent sequences:  A212619 A212620 A212621 * A212623 A212624 A212625

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Jun 01 2012

STATUS

approved

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Last modified January 17 10:59 EST 2019. Contains 319218 sequences. (Running on oeis4.)