Advances on two spectral conjectures regarding booksize of graphs

TL;DR

This paper addresses two spectral conjectures about the booksize of graphs, linking the presence of large book subgraphs to the spectral radius $\rho(G)$ and to a supersaturation phenomenon. For $B_{r+1}$-free graphs with $m$ edges and $m \ge (9r)^2$, the authors prove $\rho(G) \le \sqrt{m}$, with equality only for complete bipartite graphs, which immediately yields $bk(G) > (1/9)\sqrt{m}$ for Nosal graphs. They also show that for non-bipartite $B_{r+1}$-free graphs with $m \ge (240r)^2$, either $\rho^2 < m-1+2/(\rho-1)$ or $G \cong S^{+}_{m,s}$, establishing a robust supersaturation result and resolving a conjecture. The work highlights the extremal role of the family $S^{+}_{m,s}$ and demonstrates a gap between current lower bounds and the conjectured optimum for Nosal-type problems, with implications for Nosal-type extremal questions.

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.

Theorems & Definitions (28)

• Theorem 1.1: Nikiforov-book
• Conjecture 1.1: Li-Peng
• Theorem 1.2: L-L-Z-1
• Example 1: L-L-Z-1
• Theorem 1.3
• Corollary 1.1
• Conjecture 1.2: Liu-Lu
• Theorem 1.4
• Claim 2.1
• proof