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 A228599 The Wiener index of the graph obtained by applying Mycielski's construction to the rooted tree having Matula-Goebel 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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(2^n) = A228318(n). Indeed, the rooted tree corresponding to the Matula-Goebel 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., Prentice-Hall, NJ, 2001, p. 205. LINKS R. Balakrishnan, S. F. Raj, The Wiener number of powers of the Mycielskian, Discussiones Math. Graph Theory, 30, 2010, 489-498 (see Theorem 2.1). E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322. 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 Matula-Goebel 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)^2-8*V(n)+7-4*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

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Last modified January 22 12:13 EST 2019. Contains 319363 sequences. (Running on oeis4.)