A193153 Numbers of spanning trees in the graph join of C_n and C_n.

1296, 82944, 9150625, 1575296100, 391476713761, 132821015040000, 59042071787233536, 33317165538875522500, 23276866101199344597601, 19729668557004748392960000, 19950922411933407541569256321, 23731310247317631978185581240644

Offset: 3

Author

Eric W. Weisstein, Jul 16 2011

Keywords

• nonn

Comments

The graph join is the graph obtained by adding all possible edges between different graphs to the graph union.

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..80
• Eric Weisstein's World of Mathematics, Spanning Tree

Programs

Maple

with(LinearAlgebra):
a:= proc(n) local h, i, M;
      M:= Matrix(2*n, shape=symmetric);
      for h in [seq(seq([i, j+n], j=1..n), i=1..n),
                seq([[i, 1+(i mod n)], [n+i, n+1+(i mod n)]][], i=1..n)]
      do M[h[]]:= -1 od;
      for i to 2*n do M[i, i]:= -add(M[i, j], j=1..2*n) od;
      Determinant(DeleteColumn(DeleteRow(M, 1), 1))
    end:
seq(a(n), n=3..20);  # Alois P. Heinz, Jul 17 2011

Mathematica

a[n_] := Module[{h, i, M}, M = Array[0&, {2n, 2n}]; Do[M[[Sequence@@h]] = M[[Sequence@@Reverse[h]]] = -1, {h, Flatten[Table[{i, j+n}, {i, 1, n}, {j, 1, n}], 1] ~Join~ Flatten[Table[{{i, 1+Mod[i, n]}, {n+i, n+1 + Mod[i, n]}}, {i, 1, n}], 1]}]; For[i = 1, i <= 2n, i++, M[[i, i]] = -Sum[M[[i, j]], {j, 1, 2n}]]; Det[Rest /@ Rest[M]]];
Table[a[n], {n, 3, 20}] (* Jean-François Alcover, Feb 12 2023, after Alois P. Heinz *)

Extensions

Description corrected by Eric W. Weisstein, May 10 2017