I am trying to find an expression for inverse of the following matrix
$(L+\frac{1}{n}J)$
Where $L$ is the Laplacian of a simple, connected graph with $n$ vertices and $m$ edges, and $J=11^T$ is the all $1$ matrix. It is known that $L$ is singular, while the above matrix is non-singular. Does there exist some formula akin to Sherman–Morrison formula for this kind of inverse computation?