Nonregular graphs with a given maximum degree attaining the maximum spectral radius

Let G be a connected nonregular graph of order n with maximum degree Δ that attains the maximum spectral radius. Liu and Li (2008) [10] proposed a conjecture stating that G has a degree sequence ( Δ , … , Δ , 𝛿 ) with 𝛿 < Δ . For Δ = 3 and Δ = 4 , Liu (2024) [8] confirmed this conjecture by characterizing the structure of such graphs. Liu also proposed a modified version of the conjecture for fixed Δ and sufficiently large n , stating that the above δ satisfies 𝛿 = ⎧ { { ⎨ { { ⎩ Δ − 1 , i f Δ a n d 𝑛 a r e b o t h o d d ; 1 , i f Δ i s o d d a n d 𝑛 i s e v e n ; Δ − 2 , i f Δ i s e v e n . By applying traditional techniques in spectral graph theory and constructing certain graph operations on weighted graphs allowing loops, we fully characterize the structure of G for the cases Δ = 𝑛 − 2 with 𝑛 ≥ 5 , and Δ = 𝑛 − 3 with 𝑛 ≥ 5 9 .