Abstract

Let λ 1 be the largest eigenvalue and λ n the least eigenvalue of the adjacency matrix of a connected graph G of order n . We prove that if G is irregular with diameter D , maximum degree Δ , minimum degree δ and average degree d , then Δ - λ 1 > ( n - δ ) D + 1 Δ - d - D 2 - 1 . The inequality improves previous bounds of various authors and implies two lower bounds on λ n which improve previous bounds of Nikiforov. It also gives some fine tuning of a result of Alon and Sudakov. A similar inequality is also obtained for the Laplacian spectral radius of a connected irregular graph.

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