

A228599


The Wiener index of the graph obtained by applying Mycielski's construction to the rooted tree having MatulaGoebel number n.


1



5, 15, 33, 33, 62, 62, 59, 59, 103, 103, 103, 99, 99, 99, 156, 93, 99, 151, 93, 152, 152, 156, 151, 144, 221, 151, 215, 147, 152, 216, 156, 135, 221, 152, 217, 207, 144, 144, 216, 209, 151, 211, 147, 217, 292, 215, 216, 197, 213, 293, 217, 211
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OFFSET

1,1


COMMENTS

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.
a(2^n) = A228318(n). Indeed, the rooted tree corresponding to the MatulaGoebel number 2^n is the star graph K(1,n).
a(A007097(n)) = A228321(n). Indeed, A007097(n) for n=1,2,... yields the primeth recurrence sequence (A007097(1)=2, A007097(n+1)=A007097(n)th prime; first few terms are 2,3,5,11,31,127,709). The corresponding rooted trees are the path trees on n+1 vertices.


REFERENCES

D. B. West, Introduction to Graph Theory, 2nd ed., PrenticeHall, NJ, 2001, p. 205.


LINKS

Table of n, a(n) for n=1..52.
R. Balakrishnan, S. F. Raj, The Wiener number of powers of the Mycielskian, Discussiones Math. Graph Theory, 30, 2010, 489498 (see Theorem 2.1).
E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 23142322.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

In Balakrishnan et al. one proves that the Wiener index of the Mycielskian of a connected graph G is 6V^2  V  7E  4p(2)  p(3), where V is number of vertices of G, E is number of edges in G, and p(i) is number of pairs of vertices in G which are at distance i. For the rooted tree with MatulaGoebel number n these quantities can be found in A061775, A196050, and A196059.


MAPLE

with(numtheory): V := proc (n) local u, v: u := proc (n) options operator, arrow: op(1, factorset(n)) end proc: v := proc (n) options operator, arrow: n/u(n) end proc: if n = 1 then 1 elif isprime(n) then 1+V(pi(n)) else V(u(n))+V(v(n))1 end if end proc: WP := proc (n) local r, s, R: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: R := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*R(pi(n))+x)) else sort(expand(R(r(n))+R(s(n)))) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(WP(pi(n))+x*R(pi(n))+x)) else sort(expand(WP(r(n))+WP(s(n))+R(r(n))*R(s(n)))) end if end proc: p2 := proc (n) options operator, arrow: coeff(WP(n), x, 2) end proc: p3 := proc (n) options operator, arrow: coeff(WP(n), x, 3) end proc: a := proc (n) options operator, arrow: 6*V(n)^28*V(n)+74*p2(n)p3(n) end proc: seq(a(n), n = 1 .. 80);


CROSSREFS

Cf. A061775, A196050, A196059, A228318, A007097, A228321
Sequence in context: A073361 A155013 A134887 * A260918 A212983 A055004
Adjacent sequences: A228596 A228597 A228598 * A228600 A228601 A228602


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Aug 29 2013


STATUS

approved



