On properties of adjoint polynomials of graphs and their applications

For a graph G, we denote by P (G, λ) the chromatic polynomial of G and by h(G, x) the adjoint polynomial of G. A graph G is said to be chromatically unique if for any graph H, P (H, λ) = P (G, λ) implies H ∼= G. In this paper, we investigate some algebraic properties of the adjoint polynomials of some graphs. Using these properties, we obtain necessary and sufficient conditions for Kn − E(∪a,bT1,a,b) and (∪iCni) ∪ (∪iDmj) ∪ (∪a,bT1,a,b) to be chromatically unique if Gi ∈ {Cn, Dn, T1,a,b|n ≥ 5, 3 ≤ a ≤ 10, a ≤ b} and h(Pm) |h(Gi) for all m ≥ 2. Moreover, many new chromatically unique graphs are given. 292 ZHAO, LI, LIU AND WANG