I'm having trouble finding the eigenvalue for a given graph; but more specifically I can't seem to find the characteristic polynomial. My book tells me that the characteristic polynomial of a simple graph with n vertices is the determinant of the matrix (A-[tex]\lambda[/tex]I), where A is the adjaceny matrix and I is the n X n identity matrix. What is [tex]\lambda[/tex]? And I've read someplace else that the characteristic equation of a matrix is the determinant of (xI-A) - which is right, or are they both? And isn't the root of the equation the eigenvalue?