Abstract:

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

$$\begin{eqnarray*} &&F_n=\frac{2^{n-1}}{n}\sqrt{\prod_{k=1}^{n-1}(1-\cos\frac{k\... ...sin^2{\frac{k\pi}{n}})=L_{2n}-2=F_{2n+2}-F_{2n-2}-2,\;\;n\geq 1, \end{eqnarray*}$$

where $F_n$ and $L_n$ denote the Fibonacci and Lucas numbers, respectively. Also, we give a new proof for the identity:

$$F_n=\prod_{k=1}^{\lfloor\frac{n-1}{2}\rfloor}(1+4\sin^2{\frac... ...or\frac{n-1}{2}\rfloor}(1+4\cos^2{\frac{k\pi}{n}}),\;\;n\geq 4.$$