发表机构
Shenzhen University（深圳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究稠密情形下谱极值连通非正则图的度序列，证明Liu-Li猜想成立，即度序列由最大度数$\Delta=n-k$和尽可能大的最小度数$\delta$组成，具体形式取决于$n$和$k$的奇偶性。
AI 中文摘要
设$G$为一个具有$n$个顶点、最大度数为$\Delta$且达到最大谱半径的连通非正则图。对于$3\le\Delta\le n-2$，Liu和Li（2008）猜想$G$具有度序列$(\Delta,\ldots,\Delta,\delta)$，其中$\delta<\Delta$尽可能大。在本文中，我们研究稠密情形，即$\Delta=n-k$，其中$k\ge4$固定且$n$充分大。我们证明\\[ d(G)= \begin{cases} (\Delta,\ldots,\Delta, 1), & \text{若$n$为偶数且$k$为奇数};\\\\ (\Delta,\ldots,\Delta,\Delta-1), & \text{若$n$为奇数且$k$为偶数};\\\\ (\Delta,\ldots,\Delta,2), & \text{若$n$和$k$具有相同奇偶性}. \end{cases} \\]
英文摘要
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 $Δ=n-k$ with $k\ge4$ fixed and $n$ sufficiently large. We prove that \[ d(G)= \begin{cases} (Δ,\ldots,Δ, 1), & \text{if $n$ is even and $k$ is odd};\\ (Δ,\ldots,Δ,Δ-1), & \text{if $n$ is odd and $k$ is even};\\ (Δ,\ldots,Δ,2), & \text{if $n$ and $k$ have the same parity}. \end{cases} \]
