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

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.