A Generalization of the Characteristic Polynomial of a Graph

Given a graph G with its adjacency matrix A, the characteristic polynomial of G is defined as det(A − λI). Two graphs which have the same characteristic polynomial are called cospectral. It is known (see [2]) that there are non-isomorphic graphs which are co-spectral. In this note we consider the following generalization of the characteristic polynomial of a graph: For a graph G with adjacency matrix A, define A(x, y) as the matrix, derived from A, in which the 1s are replaced by the indeterminate x and 0s (other than the diagonals) are replaced by y. The L-polynomial of G is defined as: LG(x, y, λ) := det(A(x, y)− λI). It follows that if two graphs have the same L-polynomial, then they are co-spectral, as well as their complements are co-spectral. We show a (surprising) converse to this fact: If two graphs are co-spectral, and their complements are also co-spectral, then they have the same L-polynomial.