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

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