On the minimum real roots of the σ -polynomials and chromatic uniqueness of graphs

Let β(G) denote the minimum real root of the σ-polynomial of the complement of a graph G and δ(G) the minimum degree of G. In this paper, we give a characterization of all connected graphs G with β(G)≥-4. Using these results, we establish a sufficient and necessary condition for a graph G with p vertices and δ(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 confirmed.