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

The chromatic polynomial of a simple graph GG with n>0n>0 vertices is a polynomial ∑k=1nαk(G)x(x−1)⋯(x−k+1) of degree nn, where αk(G)αk(G) is the number of kk-independent partitions of GG for all kk. The adjoint polynomial of GG is defined to be ∑k=1nαk(G¯)xk, where G¯ is the complement of GG. 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.