The characteristic polynomial is given by

$$ p(\lambda) = det(A - \lambda I) $$

In this example we have:

$$ \begin{aligned} p(x) &= det(A - x I) = det \left( \left[ \begin{matrix}0&1&0\\1&0&1\\0&1&0\end{matrix} \right] - \left[ \begin{matrix} \lambda & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \lambda \end{matrix} \right] \right) = \\ &= \left| \begin{matrix}0 - \lambda &1&0\\1&0 - \lambda &1\\0&1&0 - \lambda \end{matrix} \right| = \\ &= (-\lambda) (-\lambda) (-\lambda) + 1 \cdot 1 \cdot 0 + 0 \cdot 1 \cdot 1 - (-\lambda) \cdot 1 \cdot 1 - 1 \cdot 1 \cdot (-\lambda) - 0 \cdot (-\lambda) \cdot 0 = \\ & = -\lambda^3 + 0 + 0 - (-\lambda) - (-\lambda) - 0 = \\ & = -\lambda^3+2\lambda \end{aligned} $$

Characteristic polynomial calculator