Adjoint polynomials of bridge–path and bridge–cycle graphs and Chebyshev polynomials

The chromatic polynomial of a simple graph G with n > 0 vertices is a polynomial ∑ k = 1 n α k ( G ) x ( x − 1 ) ⋯ ( x − k + 1 ) of degree n , where α k ( G ) is the number of k -independent partitions of G for all k . The adjoint polynomial of G is defined to be ∑ k = 1 n α k ( G ¯ ) x k , where G ¯ is the complement of G . We find explicit formulas for the adjoint polynomials of the bridge–path and bridge–cycle graphs. Consequence, we find the zeros of the adjoint polynomials of several families of graphs.