Adjoint polynomials of bridge-path and bridge-cycle graphs and Chebyshev polynomials (Q2275400)

The chromatic polynomial of a simple graph \(G\) with \(n>0\) vertices is a polynomial \[ \sum_{k=1}^{n}\alpha_k(G)x(x-1)\cdots(x-k+1) \] of degree \(n\), where \(\alpha_k(G)\) is the number of \(k\)-independent partitions of \(G\) for all \(k\). The adjoint polynomial is defined to \(\sum_{k=1}^{n}\alpha_k(\bar{G})x^k\), where \(\bar{G}\) is the complement of \(G\). In this paper, the author compute the adjoint polynomials of the bridge-path and bridge-cycle graphs. The author also compute the zeros of the adjoint polynomials of several families of graphs.