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Let G be a connected graph.We denote by σ（G，x）and δ（G）respectively the σ-polynomial and the edge-density of G,wher δ（G）=|E（G）|/（2^V(G)|).If σ（G，x）has at least an unreal root,then G is said to be a σ-unreal graph.Let δ（n) be the mionimum edge-density over