Abstract Let G and H be two graph and P(G, x) and P(H, x) are their chromatic polynomial, respectively. The two graphs G and H are said to be chromatic equivalent denoted by G ∼ H if P(G, x) = P(H, x). A graph G is called chromatically unique graph if no other graph has the chromatic polynomial as the graph G. In this paper, the chromatic uniqueness of a new family of 6-bridge graph θ(r, r, r, s, t, u), where 2 ≤ r ≤ s ≤ t ≤ u is investigated.