Let $A$ be a symmetric and invertible matrix. I know that if $A$ has a constant diagonal and a constant off-diagonal ($A=\alpha I + \beta\tilde1$, where $I$ is an identity matrix, $\tilde{1}$ is a matrix of ones, and $\alpha,\beta$ are some scalars), then $A^{-1}$ has a constant diagonal.
What are other non-obvious possible structures of $A$ that guarantee that $A^{-1}$ has a constant diagonal vector?
