dorsal/arxiv
View SchemaAdvances on two spectral conjectures regarding booksize of graphs
| Authors | Mingqing Zhai, Rui Li, Zhenzhen Lou |
|---|---|
| Categories | |
| ArXiv ID | 2601.10163vv1 |
| URL | https://arxiv.org/abs/2601.10163 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
Abstract
The \emph{booksize} $ \mathrm{bk}(G) \) of a graph $ G $, introduced by Erd\H{o}s, refers to the maximum integer $ r $ for which $G$ contains the book $ B_r $ as a subgraph. This paper investigates two open problems in spectral graph theory related to the booksize of graphs. First, we prove that for any positive integer $r$ and any $ B_{r+1} $-free graph $ G $ with $ m \geq (9r)^2 $ edges, the spectral radius satisfies $ \rho(G) \leq \sqrt{m} $. Equality holds if and only if $ G $ is a complete bipartite graph. This result improves the lower bound on the booksize of Nosal graphs (i.e., graphs with $ \rho(G) > \sqrt{m} $) from the previously established $ \mathrm{bk}(G) > \frac{1}{144}\sqrt{m} $ to $ \mathrm{bk}(G) > \frac{1}{9}\sqrt{m} $, presenting a significant advancement in the booksize conjecture proposed Li, Liu, and Zhang. Second, we show that for any positive integer $r$ and any non-bipartite $ B_{r+1} $-free graph $ G $ with $ m \geq (240r)^2 $ edges, the spectral radius $\rho$ satisfies $\rho^2<m-1+\frac{2}{\rho-1}$, unless $G$ is isomorphic to $S^+_{m,s}$ for some $s\in\{1,\ldots,r\}$. This resolves Liu and Miao's conjecture and further reveals an interesting phenomenon: even with a weaker spectral condition, $\rho^2\geq m-1+\frac2{\rho-1}$, we can still derive the supersaturation of the booksize for non-bipartite graphs.
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"abstract": "The \\emph{booksize} $ \\mathrm{bk}(G) \\) of a graph $ G $, introduced by Erd\\H{o}s, refers to the maximum integer $ r $ for which $G$ contains the book $ B_r $ as a subgraph. This paper investigates two open problems in spectral graph theory related to the booksize of graphs.\n First, we prove that for any positive integer $r$ and any $ B_{r+1} $-free graph $ G $ with $ m \\geq (9r)^2 $ edges, the spectral radius satisfies $ \\rho(G) \\leq \\sqrt{m} $. Equality holds if and only if $ G $ is a complete bipartite graph. This result improves the lower bound on the booksize of Nosal graphs (i.e., graphs with $ \\rho(G) \u003e \\sqrt{m} $) from the previously established $ \\mathrm{bk}(G) \u003e \\frac{1}{144}\\sqrt{m} $ to $ \\mathrm{bk}(G) \u003e \\frac{1}{9}\\sqrt{m} $, presenting a significant advancement in the booksize conjecture proposed Li, Liu, and Zhang.\n Second, we show that for any positive integer $r$ and any non-bipartite $ B_{r+1} $-free graph $ G $ with $ m \\geq (240r)^2 $ edges, the spectral radius $\\rho$ satisfies $\\rho^2\u003cm-1+\\frac{2}{\\rho-1}$, unless $G$ is isomorphic to $S^+_{m,s}$ for some $s\\in\\{1,\\ldots,r\\}$. This resolves Liu and Miao\u0027s conjecture and further reveals an interesting phenomenon: even with a weaker spectral condition, $\\rho^2\\geq m-1+\\frac2{\\rho-1}$, we can still derive the supersaturation of the booksize for non-bipartite graphs.",
"arxiv_id": "2601.10163",
"authors": [
"Mingqing Zhai",
"Rui Li",
"Zhenzhen Lou"
],
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"math.CO"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Advances on two spectral conjectures regarding booksize of graphs",
"url": "https://arxiv.org/abs/2601.10163",
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