New results on the adjugate matrix applied to graphs with a rotated edge

The adjugate matrix adj(G, x) of a simple graph G having adjacency matrix A is the adjugate of xI ? A. In this paper, we prove two new results on adj(G, x), one of which is a more general version of Jacobi?s theorem that is in terms of the characteristic polynomials of G and those of the vertex-deleted subgraphs G ? u, G ? v and G ? u ? v. These two results on adj(G, x) are used to present several necessary and sufficient conditions for G to keep its original characteristic polynomial after performing an edge rotation.