A Note on the Signless Laplacian Spectral Ordering of Graphs with Given Size 
Received:July 05, 2023 Revised:December 15, 2023 
Key Words:
signless Laplacian spectral radius upper bound ordering size circumference

Fund Project:Supported by the National Natural Science Foundation of China (Grant Nos.12071411; 12171222). 

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Abstract: 
For a simple undirected graph $G$ with fixed size $m\ge 2k~(k \in \mathbb{Z}^{+})$ and maximum degree $\Delta(G)\le mk$, 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 trianglefree 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: 123136]. 
Citation: 
DOI:10.3770/j.issn:20952651.2024.03.003 
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