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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.

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    $\begingroup$ To point out an obvious thing: because of (Laplacian) + (diagonalmatrix with the negated vertex degrees on the diagonal) = (-adjacencymatrix), and since the cofactors of (-adjacencymatrix) equal the cofactors of the adjacencymatrix itself, this question is at least as difficult as asking: What can appreciably be said in general about cofactors of adjacency matrices of undirected graphs? This seems difficult and not to have been done. $\endgroup$
    – Peter Heinig
    Commented Aug 18, 2017 at 17:33

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You might want to take a look at:

Bapat, R.B.; Lal, A.K.; Pati, S., A formula for all minors of the adjacency matrix and an application, Spec. Matrices 2, No. 1, 89-98 (2014). ZBL1291.05117.

which at least addresses the Peter Heinig comment.

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