Abstract

In this paper, using the number of spanning trees in some classes of graphs, we prove the identities:

F_{n} = \frac{2^{n−1}}{n} \sqrt{∏_{k=1}^{n−1}(1 − cos \frac{kπ}{n} cos \frac{3kπ}{n}}), n ≥ 2,

∏_{k=0}^{n−1}(1 + 4 sin^{2} \frac{kπ}{n}) = L_{2n} − 2 = F_{2n+2} − F_{2n−2} − 2, n ≥ 1,

where F_{n} and L_{n} denote the Fibonacci and Lucas numbers, respectively. Also, we give a new proof for the identity:

F_{n} = ∏_{k=1}^{⌊ \frac{n−1}{2} ⌋}(1 + 4 sin^{2} \frac{kπ}{n}) = ∏_{k=1}^{⌊ \frac{n−1}{2} ⌋}(1 + 4 cos^{2} \frac{kπ}{n}), n ≥ 4.