Consider the graph $G(A)$ with A as its adjacency matrix. Let $L$ be its Laplacian and $L_{sym} = D^{\frac{1}{2}}LD^{\frac{1}{2}}$ be the normalized Laplacian. Now let $A(L_{sym}) = I - L_{sym}$ denote the corresponding weighted adjacency matrix.
My question is when does $A(L_{sym})$ have a uniform degree distribution? Here the degree of node $i$ is defined as $d_i = \sum_{j} A(L_{sym})_{i,j}$.
By looking at some examples it seems that although $\{d_i\}_i$ are different, $\sum_{j} A(L_{sym})^2_{i,j}$ seems to be the same for all $i$. Is this in general correct?
It would be useful to know what can be said about $d_i$ for $A(L_{sym})$.
Thanks,