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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.

%I A228599 #13 May 28 2018 06:04:12
%S A228599 5,15,33,33,62,62,59,59,103,103,103,99,99,99,156,93,99,151,93,152,152,
%T A228599 156,151,144,221,151,215,147,152,216,156,135,221,152,217,207,144,144,
%U A228599 216,209,151,211,147,217,292,215,216,197,213,293,217,211
%N A228599 The Wiener index of the graph obtained by applying Mycielski's construction to the rooted tree having Matula-Goebel number n.
%C A228599 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.
%C A228599 a(2^n) = A228318(n). Indeed, the rooted tree corresponding to the Matula-Goebel number 2^n is the star graph K(1,n).
%C A228599 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.
%D A228599 D. B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, NJ, 2001, p. 205.
%H A228599 R. Balakrishnan, S. F. Raj, <a href="http://dx.doi.org/10.7151/dmgt.1509">The Wiener number of powers of the Mycielskian</a>, Discussiones Math. Graph Theory, 30, 2010, 489-498 (see Theorem 2.1).
%H A228599 E. Deutsch, <a href="http://dx.doi.org/10.1016/j.dam.2012.05.012">Rooted tree statistics from Matula numbers</a>, Discrete Appl. Math., 160, 2012, 2314-2322.
%H A228599 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A228599 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.
%p A228599 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);
%Y A228599 Cf. A061775, A196050, A196059, A228318, A007097, A228321
%K A228599 nonn
%O A228599 1,1
%A A228599 _Emeric Deutsch_, Aug 29 2013