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).
Author NameAffiliation
Nannan LIU School of Mathematics and Statistics, Qinghai Normal University, Qinghai 810008, P. R. China
School of Mathematics and Statistics, Yancheng Teachers University, Jiangsu 224002, P. R. China 
Shuguang GUO School of Mathematics and Statistics, Yancheng Teachers University, Jiangsu 224002, P. R. China 
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Abstract:
      For a simple undirected graph $G$ with fixed size $m\ge 2k~(k \in \mathbb{Z}^{+})$ and maximum degree $\Delta(G)\le m-k$, 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].
Citation:
DOI:10.3770/j.issn:2095-2651.2024.03.003
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