Two remarks on the adjoint polynomial

One can define the adjoint polynomial of the graph G as follows. Let a k ( G ) denote the number of ways one can cover all vertices of the graph G by exactly k disjoint cliques of G . Then the adjoint polynomial of G is h ( G , x ) = ∑ k = 1 n ( − 1 ) n − k a k ( G ) x k , where n denotes the number of vertices of the graph G . In this paper we show that the largest real root γ ( G ) of h ( G , x ) has the largest absolute value among the roots. We also prove that γ ( G ) ≤ 4 ( Δ − 1 ) , where Δ denotes the largest degree of the graph G . This bound is sharp.