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Ordering Graphs with Given Size by Their Signless Laplacian Spectral Radii

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Abstract

Let q(G) denote the signless Laplacian spectral radius of a graph G. In this paper, we first give an upper bound on q(G) of a connected graph G with fixed size \(m\ge 3k(k \in {\mathbb {Z}}^{+})\) and maximum degree \(\Delta \le m-k\). For two connected graphs \(G_1\) and \(G_2\) with size \(m\ge 4\), employing this upper bound, we prove that \(q(G_1)>q(G_2)\) if \(\Delta (G_1)>\Delta (G_2)\) and \(\Delta (G_1)\ge \frac{2m}{3}+1\). As an application, we determine the first \(\lfloor d/2\rfloor \) graphs with the largest signless Laplacian spectral radius among all graphs with fixed size and diameter.

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Acknowledgements

The authors are grateful to the anonymous referees for valuable suggestions and corrections which result in an improvement of the original manuscript.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, 224002, Jiangsu, People’s Republic of China

    Rong Zhang & Shu-Guang Guo

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  1. Rong Zhang
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  2. Shu-Guang Guo
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Correspondence to Shu-Guang Guo.

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Communicated by Sandi Klavžar.

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Supported by the National Natural Science Foundation of China (Nos. 12071411, 12171222)

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Zhang, R., Guo, SG. Ordering Graphs with Given Size by Their Signless Laplacian Spectral Radii. Bull. Malays. Math. Sci. Soc. 45, 2165–2174 (2022). https://doi.org/10.1007/s40840-022-01312-1

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  • DOI: https://doi.org/10.1007/s40840-022-01312-1

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