New bounds for chromatic polynomials and chromatic roots

If G is a k-chromatic graph of order n then it is known that the chromatic polynomial of G, π(G,x), is at most x(x−1)⋯(x−(k−1))xn−k=(x)↓kxn−k for every x∈N. We improve here this bound by showing that π(G,x)≤(x)↓k(x−1)Δ(G)−k+1xn−1−Δ(G) for every x∈N, where Δ(G) is the maximum degree of G. Secondly, we show that if G is a connected k-chromatic graph of order n where k≥4 then π(G,x) is at most (x)↓k(x−1)n−k for every real x≥n−2+(n2−k2−n+k)2 (it had been previously conjectured that this inequality holds for all x≥k). Finally, we provide an upper bound on the moduli of the chromatic roots that is an improvement over known bounds for dense graphs.