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A193402 The Matula-Göbel numbers of the rooted trees such that 2 is an eigenvalue of the Laplacian matrix.

%I A193402 #22 Mar 07 2017 06:18:28
%S A193402 2,5,6,15,18,22,23,26,31,41,45,54,55,65,66,69,78,93,94,103,122,123,
%T A193402 135,137,158,162,165,166,167,195,198,202,207,211,234,235,242,253,254,
%U A193402 279,282,283,286,299,305,309,338,341,343,358,366,369,394,395,401,403
%N A193402 The Matula-Göbel numbers of the rooted trees such that 2 is an eigenvalue of the Laplacian matrix.
%C A193402 The Matula-Göbel 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-Göbel 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-Göbel numbers of the m branches of T.
%H A193402 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H A193402 Yi-zheng Fan, <a href="http://dx.doi.org/10.1007/s10255-004-0166-y">On the eigenvalue two and matching number of a tree</a>, Acta Math. Appl. Sinica, English Series, 20, 2004, 257-262.
%H A193402 F. Göbel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H A193402 I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H A193402 I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H A193402 D. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H A193402 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A193402 Let T be a rooted tree with root b. If b has degree 1, then let A be the rooted tree with root c, obtained from T by deleting the edge bc emanating from the root. If b has degree >=2, then A is obtained (not necessarily in a unique way) by joining at b two trees B and C, rooted at b. It is straightforward to express the distance matrix of T in terms of the entries of the distance matrix of A (resp. of B and C). Making use of this, the Maple program (improvable!) finds recursively the distance matrices of the rooted trees with Matula-Göbel numbers 1..1000 (upper limit can be altered), then switches (easily) to the Laplacian matrices and finds their  characteristic polynomials.
%e A193402 5 is in the sequence because the rooted tree with Matula-Göbel number 5 is the path on 4 vertices; the Laplacian matrix is [1,-1,0,0; -1,2,-1,0; 0,-1,2,-1;0,0,-1,1] with characteristic polynomial x(x-2)(x^2 -4x +2).
%p A193402 with(numtheory): with(linalg): with(LinearAlgebra): DA := proc (d) local aa: aa := proc (i, j) if d[i, j] = 1 then 1 else 0 end if end proc: Matrix(RowDimension(d), RowDimension(d), aa) end proc: AL := proc (a) local ll: ll := proc (i, j) if i = j then add(a[i, k], k = 1 .. RowDimension(a)) else -a[i, j] end if end proc: Matrix(RowDimension(a), RowDimension(a), ll) end proc: V := 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+V(pi(n)) else V(r(n))+V(s(n))-1 end if end proc: d := proc (n) local r, s, C, a: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: C := proc (A, B) local c: c := proc (i, j) options operator, arrow: A[1, i]+B[1, j+1] end proc: Matrix(RowDimension(A), RowDimension(B)-1, c) end proc: a := proc (i, j) if i = 1 and j = 1 then 0 elif 2 <= i and 2 <= j then dd[pi(n)][i-1, j-1] elif i = 1 then 1+dd[pi(n)][1, j-1] elif j = 1 then 1+dd[pi(n)][i-1, 1] else  end if end proc: if n = 1 then Matrix(1, 1, [0]) elif bigomega(n) = 1 then Matrix(V(n), V(n), a) else Matrix(blockmatrix(2, 2, [dd[r(n)], C(dd[r(n)], dd[s(n)]), Transpose(C(dd[r(n)], dd[s(n)])), SubMatrix(dd[s(n)], 2 .. RowDimension(dd[s(n)]), 2 .. RowDimension(dd[s(n)]))])) end if end proc: for n to 1000 do dd[n] := d(n) end do: S := {}: for n to 500 do if subs(x = 2, CharacteristicPolynomial(AL(DA(d(n))), x)) = 0 then S := `union`(S, {n}) else  end if end do: S;
%Y A193402 Cf. A061775, A184187, A193403, A193401.
%K A193402 nonn
%O A193402 1,1
%A A193402 _Emeric Deutsch_, Feb 11 2012