The question is how to find a closed formula for the characteristic polynomial and the eigenvalues of the $N$*$N$ adjacency matrix $A$ of a fully connected network of size $N$:
$$ A =\begin{pmatrix} 0 & 1 & 1 & \cdots & 1 \\ 1 & 0 & 1 & \cdots & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \cdots & 0 \end{pmatrix}.$$
My attempt:
I tried to find a recursive formula for the characteristic polynomial $P(\lambda)$. It seems that: $$P(\lambda)_{N+1} = (-\lambda)P(\lambda)_{N} - NQ_N(\lambda)$$ where $Q_N(\lambda)$ is the determinant of the following matrix:
$$ \begin{pmatrix} 1 & 1 & 1 & \cdots & 1 \\ 1 & -\lambda & 1 & \cdots & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \cdots & -\lambda \end{pmatrix}.$$
Again trying to find a recursion formula for $Q_N$, using gaussians elimination I found that: $$Q_N(\lambda) = (-1)^{N-1}(\lambda+1)^{N-1}$$ and so: $$P(\lambda)_{N+1} = (-\lambda)P(\lambda)_{N} +(-1)^{N}N(\lambda+1)^{N-1}$$ but at this point I couldn't go on.
Looking for a pattern, starting from small N, it seems that the relation I'm looking for is: $$P_N(\lambda) = (\lambda -N + 1)(\lambda+1)^{N-1}$$ but I'm not sure and it seems in discord with the previous recurrence relation.