A note on eigenvalue bounds for independence numbers of non-regular graphs

Let GG be a simple connected graph of order n≥2n≥2 with maximum degree ΔΔ and minimum degree δδ, and eigenvalues λ1≥λ2≥⋯≥λnλ1≥λ2≥⋯≥λn. It is shown that the independence number of GG can be bounded from above by Δ−δ−λnΔn and λ1−λn+Δ−2δλ1−λn+Δ−δn.