Advances on two spectral conjectures regarding booksize of graphs

Abstract

The 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.