Let be a graph attaining the maximum spectral radius among all connected nonregular graphs of order with maximum degree Δ. Let be the spectral radius of . A nice conjecture due to Liu et al. (2007) asserts that for each fixed Δ. Concerning an important structural property of the extremal graphs , Liu and Li (2008) put forward another conjecture which states that has exactly one vertex of degree strictly less than Δ. In this paper, we make progress on the two conjectures. To be precise, we disprove the first conjecture for all by showing that For small Δ, we determine the precise asymptotic behavior of . In particular, we show that if ; and if . We also confirm the second conjecture for and by determining the precise structure of extremal graphs. Furthermore, we show that the extremal graphs for must have a path-like structure built from specific blocks.

中文翻译：

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规定最大次数的非正则图的极值谱半径


令 为在所有具有最大阶次 Δ 的连通非正则图中获得最大谱半径的图。设 为 的谱半径。 Liu 等人提出了一个很好的猜想。 (2007) 断言对于每个固定的 Δ。针对极值图的一个重要结构性质，Liu和Li(2008)提出了另一个猜想，即恰好有一个度数严格小于Δ的顶点。在本文中，我们对这两个猜想都取得了进展。准确地说，我们通过证明对于较小的 Δ，我们确定 的精确渐近行为来反驳所有人的第一个猜想。特别是，我们证明如果；如果 .我们还通过确定极值图的精确结构来证实第二个猜想。此外，我们表明 的极值图必须具有由特定块构建的类似路径的结构。