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

Abstract

Let $G$ be an $n$-vertex connected nonregular graph with maximum degree $\Delta$ that attains the maximum spectral radius. For $3\le \Delta \le n-2$, Liu and Li (2008) conjectured that $G$ has the degree sequence $(\Delta ,\dots ,\Delta ,\delta )$ with $\delta <\Delta$ as large as possible. In this paper, we study the dense case where $\Delta =n-k$ with $k\ge 4$ fixed and $n$ sufficiently large. We prove that \[ d(G)= \begin{cases} (Δ,\ldots,Δ, 1), & \text{if $n$ is even and $k$ is odd};\\ (Δ,\ldots,Δ,Δ-1), & \text{if $n$ is odd and $k$ is even};\\ (Δ,\ldots,Δ,2), & \text{if $n$ and $k$ have the same parity}. \end{cases} \]