OFFSET
0,3
COMMENTS
For Laguerre graphs (open and closed ones) see the W. Lang link on Jacobi graphs under A201198. There one also finds a sketch of the closed Laguerre graph Lc_4 as Fig.4.
The total number of round trips on the closed Laguerre graph Lc_N, for N>=3, with N vertices N^2 loops, binomial(N,2) lines between neighboring vertices and two lines between the first and the last vertex (in total (3*N-1)*N/2+2 = (3*N^2-N+4)/2 lines) is w(N,L) = sum(w(N,L;p_n->p_n),n=1..N) = Trace((L_N)^L) = sum((x_n^{(N)})^L,n=1..N), with the N x N symmetric adjacency matrix, also called Lc_N, having non-vanishing elements (Lc_N)[n,n] = 2*n-1, n=1..N, (Lc_N)[n,n+1] = (Lc_N)[n+1,n] = n, n=1..N-1, and (Lc_N)[1,N]= 2=(Lc_N)[N,1]. The eigenvalues of Lc_N are x_n^{(N)}. They are the zeros of the characteristic polynomial Lac_N(x):=Det(x*1_N -Lc_N) with the N x N unit matrix 1_N. These are the polynomials Lac_N(x) = La(N,x) - 4*La1(N-2,x) - 4*(N-1)!, with the ordinary monic Laguerre polynomials La(N,x) with coefficient array given by A021009(n,m)*(-1)^n and the first associated monic Laguerre polynomials La1(N-2,x) with coefficient array given by A199577(n,m). For N=1 one has Lc_1=L_1 (Laguerre graph with one vertex and one loop) with L_1(x)=x-1, and for N=2 one has a graph where one vertex has one loop, the other three, and there are two lines joining these vertices, hence Lc_2(x)= x^2-4*x-1.
LINKS
FORMULA
a(K,N) = w(N,K-N+1),K>=0, N=1,...,K+1, with w(N,L) the total number of round trips of length L on the closed Laguerre graph Lc_N described above in the comment section.
The o.g.f. of w(N,L) is: G(N,x)=y*(d/dx)Lac_N(x)/Lac_N(x) with y=1/x.
The characteristic polynomial Lac_N(x) has also been given in the comment section above.
EXAMPLE
The array w(N,L) starts:
N\L 0 1 2 3 4 5 6 ...
1: 1 1 1 1 1 1 1
2: 2 4 12 40 136 464 1584
3: 3 9 53 357 2489 17509 123449
4: 4 16 120 1024 9424 89536 862560
5: 5 25 233 2545 29985 367505 4599521
6: 6 36 404 5400 78392 1188336 18460016
7: 7 49 645 10213 176473 3195829 59473593
8: 8 64 968 17728 355536 7493504 162671840
9: 9 81 1385 28809 657953 15826041 392792273
...The triangle a(K,N) = w(N,K-N+1) starts:
K\N 1 2 3 4 5 6 7 8 9..
0: 1
1: 1 2
2: 1 4 3
3: 1 18 9 4
4: 1 76 53 16 5
5: 1 322 357 120 25 6
6: 1 1364 2489 1024 233 36 7
7: 1 5778 17509 9424 2545 404 49 8
8: 1 24476 123449 89536 29985 5400 645 64 9
...
For the graph Lc_4, shown in the W. Lang link as Figure 4, the counting for round trips of length L=2 for each of the four vertices V_i, i=1..4, read from left to right, is as follows.
V_1: 1+1+(1+1+2*1), V_2: 3+2*binomial(3,2)+1+(1+1+2*1),
V_3: 5+2*binomial(5,2)+(1+1+2*1)+(3+2*binomial(3,2)),
V_4: 7+2*binomial(7,2)+(3+2*binomial(3,2))+(1+1+2*1),
this sums to the total number w(4,2)= 120 = a(5,4).
Compared to the open L_4 graph (see the corresponding A201198 entry 4*28 = 112) one has to add 2*(1+1+2*1)=8 from the new two lines joining V_1 and V_4.
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Nov 30 2011
STATUS
approved