谱极值与极图是当今图谱理论研究的热点问题, 学者们非常关注研究图的谱半径达到最大或者最小值所对应的极图。本文刻画了直径为$4$的超树的第二大无符号拉普拉斯谱半径的极图。设$\mathbb{S}(m, 4, k)$是指有$m$条边直径为4的$k$一致超树的集合, $S_3(m, 4, k)$是由直径为4的疏松路$v_1e_1v_2e_2v_3e_3v_4e_4v_5$在顶点$v_4$处悬挂$m-4$条边得到的$k$一致超树。本文首先介绍了超图中边扰动的定义以及相关定理。然后, 根据边扰动等方法, 研究得出$S_3(m, 4, k)$是$\mathbb{S}(m, 4, k)$中无符号拉普拉斯谱半径达到第二大的图。

The spectral extremal problem and graph are hot issues in the study of graph theory nowadays. Scholars are keen to study the extremal graphs attaining the maximum or minimum spectral radius of graph classes. In this paper, the extremal graph of the second largest unsigned Laplacian spectral radius of a supertree with diameter of $4$ is characterized. Let $\mathbb{S}(m, 4, k)$ be the set of $k$-uniform supertree with $m$ edges and diameter $4$, and $S_3(m, 4, k)$ be the $k$-uniform supertree obtained from a loose path $v_1e_1v_2e_2v_3e_3v_4e_4v_5$ with length $4$ by attaching $m-4$ edges at vertex $v_4$. In this paper, firstly, introducing the definition of edge-shifting operation and related theorems. Then, according to edge-shifting operation, we find $S_3(m, 4, k)$ is the graph with the second largest signless Laplacian spectral radius in $\mathbb{S}(m, 4, k)$.