A characteristic polynomial of a matrix is the left hand side of the characteristic equation (det(A – λ ⋅ I)=0). For a 2x2 matrix A, the characteristic polynomial is x2 – (a11 + a22)x +
a11a22 – a12a21, or
x2 – tr A⋅x + det A. Explicit formulae for characteristic polynomials of bigger matrices exist, but are much uglier. They are, however, all monic. Similar matrices have the same characteristic polynomials, but two matrices with the same characteristic polynomial need not be similar. A matrix and its transpose share characteristic polynomials.
According to the Cayley-Hamilton Theorem, if a matrix A has the characteristic polynomial f(x), then f(A)=0. By this fact, we know that the characteristic polynomial of A is divisible by the minimal polynomial of A. Also, f(x) can be factored into linear factors if and only if A is similar to a matrix in Jordan normal form.
The characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. Any two graphs that have the same characteristic polynomial are called cospectral (or isospectral). It is possible for two nonisomorphic graphs to have the same characteristic polynomial.
Eric W. Weisstein. "Characteristic Polynomial." From MathWorld--A Wolfram Web Resource.