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Let ÿ(G) denote the minimum real root of the -polynomial of the complement of a graph G and (G) the minimum degree of G. In this paper, we give a characterization of all connected graphs G with ÿ(G) ¿ − 4. Using these results, we establish a su cient and necessary condition for a graph G with p vertices and (G) ¿ p − 3, to be chromatically unique. Many previously known results are generalized. As a byproduct, a problem of Du (Discrete Math. 162 (1996) 109-125) and a conjecture of Liu (Discrete Math. 172 (1997) 85-92) are conÿrmed.

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