Degree sequences of spectrally extremal connected nonregular graphs with prescribed maximum degree: the dense case

Let $G$ be an $n$-vertex connected nonregular graph with maximum degree $Δ$ that attains the maximum spectral radius. For $3\leΔ\le n-2$, Liu and Li (2008) conjectured that $G$ has the degree sequence $(Δ,\ldots,Δ,δ)$ with $δ<Δ$ as large as possible. In this paper, we study the dense case where $Δ=n