For Δ=n-2 and Δ=n-3, the connected nonregular graphs of order n with maximum spectral radius are fully characterized: G(n,t) for Δ=n-2, and H1(n) or H2(n) for Δ=n-3.
Haemers, Interlacing eigenvalues and graphs, Linear Algebra Appl
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Nonregular graphs with a given maximum degree attaining maximum spectral radius
For Δ=n-2 and Δ=n-3, the connected nonregular graphs of order n with maximum spectral radius are fully characterized: G(n,t) for Δ=n-2, and H1(n) or H2(n) for Δ=n-3.