A supertree is a connected and acyclic hypergraph. Let ${\mathcal{T}}_m^{\mathrm{△},\mathrm{△}}$ be the set of the r-uniform supertrees with m edges and exactly two vertices of maximum degree ▵, where $m\ge 5$ and $3\le \mathrm{△}\le \left[\right(m+1\left)/2\right]$. We investigate the supertree with the minimum spectral radius among ${\mathcal{T}}_m^{\mathrm{△},\mathrm{△}}$. Several useful grafting operations are introduced for comparing the spectral radii of supertrees. The supertree with the minimum spectral radius among ${\mathcal{T}}_m^{\mathrm{△},\mathrm{△}}$ is characterized. In addition, a new simple method is demonstrated to obtain the supertree with the minimum spectral radius among the set of r-uniform supertrees with m edges.

中文翻译：

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具有最大度数的两个顶点的 r 一致超树的最小谱半径

超级树是一个连通的非循环超图。让${\mathcal{吨}}_{米}^{\mathrm{△},\mathrm{△}}$是具有m条边和恰好两个最大度数为 ▵ 的顶点的r一致超树的集合，其中$米\ge 5$和$3\le \mathrm{△}\le \left[\right(米+1\left)/2\right]$. 我们研究了具有最小光谱半径的超树${\mathcal{吨}}_{米}^{\mathrm{△},\mathrm{△}}$. 介绍了几种有用的嫁接操作来比较超级树的光谱半径。具有最小光谱半径的超树${\mathcal{吨}}_{米}^{\mathrm{△},\mathrm{△}}$有特点。此外，还展示了一种新的简单方法，可以在具有m条边的r一致超树集合中获得具有最小光谱半径的超树。