we have a simple graph that it's vertices are {v_1, v_2, ... v_n}.

The Adjacency matrix  of this graph is A= (a_ij) so that;

(a_ij)=1    if      i+j belongs to the Fibonacci sequence.
(a_ij)=0    if      i+j doesn't belong  to the Fibonacci sequence.

We claim that the determinant of this matrix when n is odd is 0 and when n is even is 1 or -1 or 0.
How can we prove this claim?