AI 中文总结
该研究针对非二部图的谱书规模问题，渐近确定了Zhai等人提出的问题中的最优常数为1/4，给出了对应谱条件下的书规模下界，并构造反例证明该常数的最优性。
AI 中文摘要
图$G$的$\text{bk}(G)$是共享同一条边的三角形的最大数量。受Erdős经典猜想的推动，书规模的谱下界受到了广泛关注。对于$m-1$的正除数$s$且满足$\frac{m-1}{s}\ge2$，令$S_{m,s}^{+}$是在$K_{s,\frac{m-1}{s}}$的阶为$\frac{m-1}{s}$的部内添加一条边得到的图。Zhai等人证明，除了这个显式构造的图族外，所有满足$\rho(G)^2\ge m-1+\frac{2}{\rho(G)-1}$的$m$条边的非二部图的书规模都大于$\frac{1}{240}\sqrt{m}$，并提出了寻找最优常数的问题。\n我们渐近地回答了这个问题。对于任意$0<\varepsilon<\frac{1}{4}$，当$m$足够大时，所有满足相同谱条件、无孤立顶点的$m$边非二部图$G$，要么同构于某个对应整数$s$对应的$S_{m,s}^{+}$，要么满足$\text{bk}(G)>\left(\frac{1}{4}-\varepsilon\right)\sqrt{m}$。我们还构造了无穷多个不属于例外图族的图，证明不存在大于$\frac{1}{4}$的常数满足条件。因此$\frac{1}{4}$是Zhai等人所提问题中的最优渐近常数。
英文摘要
The $\text{bk}(G)$ of a graph $G$ is the maximum number of triangles sharing a common edge. Motivated by a classical conjecture of Erdős, spectral lower bounds for the booksize have received considerable attention. For a positive divisor $s$ of $m-1$ with $\frac{m-1}{s}\ge2$, let $S_{m,s}^{+}$ be obtained from $K_{s,\frac{m-1}{s}}$ by adding one edge inside the part of order $\frac{m-1}{s}$. Zhai et al. proved that, apart from this explicit family, every $m$-edge non-bipartite graph satisfying $ρ(G)^2\ge m-1+\frac{2}{ρ(G)-1}$ has booksize greater than $\frac{1}{240}\sqrt{m}$, and they asked for the best possible constant. We answer this question asymptotically. For every $0<\varepsilon<\frac{1}{4}$ and all sufficiently large $m$, every $m$-edge non-bipartite graph $G$ without isolated vertices satisfying the same spectral condition either is isomorphic to $S_{m,s}^{+}$ for some such integer $s$, or satisfies $\text{bk}(G)>\left(\frac{1}{4}-\varepsilon\right)\sqrt{m}$. We also give infinitely many graphs outside the exceptional family showing that no constant larger than $\frac{1}{4}$ is possible. Thus $\frac{1}{4}$ is the optimal asymptotic constant in the problem of Zhai et al.
CommentsPages:21, 0 figures, 0 tables. This paper resolves the open question on spectral booksize of non-bipartite graphs, and verifies the optimal asymptotic constant is 1/4
