login
The overall second Zagreb index of the rooted tree having Matula-Goebel number n.
10

%I #19 Mar 07 2017 11:25:27

%S 0,1,6,6,19,19,24,24,44,44,44,59,59,59,85,80,59,125,80,114,114,85,125,

%T 173,146,125,246,156,114,219,85,240,146,114,193,344,173,173,219,302,

%U 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

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

%C 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.

%C 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.

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

%D F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

%D I. Gutman and Y-N. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.

%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288.

%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

%F 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.

%e 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.

%p 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);

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

%K nonn

%O 1,3

%A _Emeric Deutsch_, Jun 01 2012