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The spectral radius of connected non-regular graphs is considered. Let λ1 be the largest eigenvalue of the adjacency matrix of a graph G on n vertices with maximum degree Δ. By studying the λ1-extremal graphs, it is proved that if G is non-regular and connected, then Δ − λ1 > Δ+1/n(3n+Δ− 8). This improves the recent results by B.L. Liu et al.