Diagonally forced systems and the spectral signature of matrix cycles

Given any square matrix, $\mathbf{M}$, whose diagonal elements are negative, and which are multiplied by a variable, $σ$, we wish to find the minimal $σ$ such that the eigenvalue of $\mathbf{M}_σ$ is exactly zero. By Gershgorin, we know that $\mathbf{M}_σ$ can be made stable by making $σ$ large enou