AI 中文总结
研究给定平均度的图的谱半径下界，通过问题分数松弛的精确解得到改进下界，肯定了Hong的猜想，刻画了离散实现条件，证明固定顶点数时有效边配置数量至少线性增长，平均渐近阶为Θ(n log n)。
AI 中文摘要
这项工作为给定平均度的图的谱半径建立了一个改进的下界。新的界来自于该问题分数松弛的精确解。我们的发现对Hong（1993）关于具有特定平均度的图的猜想给出了肯定答案，因为达到我们界的极值图被证明其最小度和最大度相差至多为1。此外，我们精确刻画了允许这种离散实现的条件。我们证明，对于固定数量的顶点n，有效边配置的数量至少随n线性增长，平均渐近阶为Θ(n log n)。
英文摘要
This work establishes an improved lower bound for the spectral radius of a graph given its average degree. The new bound follows from an exact solution of the fractional relaxation of the problem. Our findings lead to an affirmative answer to a conjecture by Hong (1993) for graphs with specific average degrees -- as the extremal graphs that meet our bound are proven to have a minimal and maximal degree that differ by at most one. Furthermore, we provide an exact characterization of the conditions that permit such discrete realizations. We prove that for a fixed number of vertices $n$, the number of valid edge configurations grows at least linearly with $n$, achieving an average asymptotic order of $Θ(n\log n)$.
Comments25 pages, 3 figures
