Advances on two spectral conjectures regarding booksize of graphs

Abstract: The \emph{booksize} $ \mathrm{bk}(G) ) of a graph $ G $, introduced by Erdős, 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 $ ρ(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 $ ρ(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 $ρ$ satisfies $ρ^2<m-1+\frac{2}{ρ-1}$, 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, $ρ^2\geq m-1+\frac2{ρ-1}$, we can still derive the supersaturation of the booksize for non-bipartite graphs.