We know (e.g. [Godsil, Royle: Algebraic Graph Theory, Lemma 13.2.3]) that any cofactor of the Laplacian matrix of a graph is constant, and is equal to the number of spanning trees of the graph. How do the cofactors change if I just add a diagonal matrix to the Laplacian matrix?

Any help would be greatly appreciated.

is at least as difficult as asking: What can appreciably be said in general about cofactors of adjacency matrices of undirected graphs?Thisseemsdifficult and not to have been done. $\endgroup$