关于边数给定图按无符号拉普拉斯谱半径排序的注记

设$k$是正整数, $G$是一个边数给定的简单无向图, 其边数$m\ge 2k$, 最大度$\Delta(G)\le m-k$, 本文给出了图$G$的无符号拉普拉斯谱半径$q(G)$的一个上界. 对边数为$m\ge 8$的两个连通图$G_1$和$G_2$, 利用这个上界我们证明了一个排序定理: 如果$\Delta(G_1)>\Delta(G_2)+1$ 且 $\Delta(G_1)\ge \frac{m}{2}+2$, 那么$q(G_1)>q(G_2)$. 对于不含三角形的图, 我们得到两个更强的结果. 作为上述排序定理的一个应用, 我们完全刻画了无符号拉普拉斯谱半径最大的围长为$c$的$m$边图, 其中$m\ge \max\{ 2c, c+9\}$, 部分解决了陈雯雯等人在Linear Algebra Appl. 645(2022)123-136]上提出的一个公开问题.

A Note on the Signless Laplacian Spectral Ordering of Graphs with Given Size

For a simple undirected graph $G$ with fixed size $m\ge 2k~(k \in \mathbb{Z}^{+})$ and maximum degree $\Delta(G)\le m-k$, we give an upper bound on the signless Laplacian spectral radius $q(G)$ of $G$. For two connected graphs $G_1$ and $G_2$ with size $m\ge 8$, employing this upper bound, we prove that $q(G_1)>q(G_2)$ if $\Delta(G_1)>\Delta(G_2)+1$ and $\Delta(G_1)\ge \frac{m}{2}+2$. For triangle-free graphs, we prove two stronger results. As an application, we completely characterize the graph with maximal signless Laplacian spectral radius among all graphs with size $m$ and circumference $c$ for $m\ge \max\{ 2c, c+9\}$, which partially answers the question proposed by Chen et al. in Linear Algebra Appl., 2022, 645: 123--136].