Is there a name for square matrices whose rows and columns (and only these; indeed, the main diagonal entries always sum to $n \alpha,$ where $n$ is the size of the matrix, and $\alpha$ each entry of the form $a_{ii}$) sum to the same value?
For example, something like $$\begin{bmatrix} 2&1&0&1\\ 1&2&1&0\\ 0&1&2&1\\ 1&0&1&2 \end{bmatrix}.$$
PS. As it might help, an additional property that these matrices have is that they are at least persymmetric (that is in addition to the constancy of the row/column-sum above), although the antidiagonal need not be constant.