A196054 -id:A196054 - cp's OEIS frontend

A212622 The overall second Zagreb index of the rooted tree having Matula-Goebel number n.

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

Offset: 1

Emeric Deutsch, Jun 01 2012

nonn

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.

			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.

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.

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

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

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);

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.

A196053 The first Zagreb index of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 2, 6, 6, 10, 10, 12, 12, 14, 14, 14, 16, 16, 16, 18, 20, 16, 20, 20, 20, 20, 18, 20, 24, 22, 20, 24, 22, 20, 24, 18, 30, 22, 20, 24, 28, 24, 24, 24, 28, 20, 26, 22, 24, 28, 24, 24, 34, 26, 28, 24, 26, 30, 32, 26, 30, 28, 24, 20, 32, 28, 22, 30, 42, 28, 28, 24, 26, 28, 30, 28, 38, 26, 28, 32, 30, 28, 30, 24, 38, 36, 24, 24, 34, 28, 26, 28, 32, 34, 36, 30, 30, 26, 28, 32, 46, 28, 32, 32, 36

Offset: 1

Emeric Deutsch, Sep 28 2011

nonn

The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices.

			a(7)=12 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1+9+1+1=12).
a(2^m) = m(m+1) because the rooted tree with Matula-Goebel number 2^m is a star with m edges.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
Ivan Gutman and Kinkar C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
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 Rev. 10 (1968) 273.
S. Nikolic, G. Kovacevic, A. Milicevic, and N. Trinajstic, The Zagreb indices 30 years after, Croatica Chemica Acta, 76, 2003, 113-124.
Index entries for sequences related to Matula-Goebel numbers

Cf. A196054.

Cf. A049084, A020639, A001222.

import Data.List (genericIndex)
a196053 n = genericIndex a196053_list (n - 1)
a196053_list = 0 : g 2 where
   g x = y : g (x + 1) where
     y | t > 0     = a196053 t + 2 + 2 * a001222 t
       | otherwise = a196053 r + a196053 s -
                     a001222 r ^ 2  - a001222 s ^ 2 + a001222 x ^ 2
       where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013

with(numtheory): a := 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 a(pi(n))+2+2*bigomega(pi(n)) else a(r(n))+a(s(n))-bigomega(r(n))^2-bigomega(s(n))^2+bigomega(n)^2 end if end proc: seq(a(n), n = 1 .. 100);

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
a[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, a[PrimePi[n]] + 2 + 2*PrimeOmega[PrimePi[n]], True, a[r[n]] + a[s[n]] - PrimeOmega[r[n]]^2 - PrimeOmega[s[n]]^2 + PrimeOmega[n]^2];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 25 2024, after Maple code *)

a(1)=0; if n = prime(t) (the t-th prime), then a(n)=a(t)+2+2G(t); if n=r*s (r,s>=2), then a(n)=a(r)+a(s)-G(r)^2-G(s)^2+G(n)^2; G(m) is the number of prime factors of m, counted with multiplicities. The Maple program is based on this recursive formula.

cp's OEIS Frontend

A212622 The overall second Zagreb index of the rooted tree having Matula-Goebel number n.

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