|
|
A196054
|
|
The second Zagreb index of the rooted tree with Matula-Goebel number n.
|
|
2
|
|
|
0, 1, 4, 4, 8, 8, 9, 9, 12, 12, 12, 14, 14, 14, 16, 16, 14, 19, 16, 18, 18, 16, 19, 22, 20, 19, 24, 21, 18, 23, 16, 25, 20, 18, 22, 28, 22, 22, 23, 26, 19, 26, 21, 22, 28, 24, 23, 32, 24, 27, 22, 26, 25, 34, 24, 30, 26, 23, 18, 32, 28, 20, 31, 36, 27, 27, 22, 24, 28, 30, 26, 39, 26, 28, 32, 30, 26, 31, 22, 36, 40, 23, 24, 36, 26, 26, 27, 30, 32, 38
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
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.
|
|
LINKS
|
|
|
FORMULA
|
a(1)=0; if n=p(t) (the t-th prime), then a(n) = a(t)+b(t)+G(t)+1; if n=rs (r,s>=2), then a(n)=a(r)+a(s)+b(r)G(s)+b(s)G(r); here b(m) is the sum of the degrees of the nodes at level 1 of the rooted tree having Matula-Goebel number m and G(m) is the number of prime factors of m, counted with multiplicities. The Maple program is based on this recursive formula.
|
|
EXAMPLE
|
a(7)=9 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1*3+3*1+3*1=9).
a(2^m) = m^2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
|
|
MAPLE
|
with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+bigomega(pi(n)) else b(r(n))+b(s(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then a(pi(n))+b(pi(n))+bigomega(pi(n))+1 else a(r(n))+a(s(n))+b(r(n))*bigomega(s(n))+b(s(n))*bigomega(r(n)) end if end proc: seq(a(n), n = 1 .. 90);
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|