AI 中文总结
该研究解决固定直径下的渐近度-直径问题，证明Bollobás猜想，通过正则图$H_{k,q}$得到下界，并构造满足线图直径条件的图，渐近达到摩尔界。
AI 中文摘要
对于正整数$d$和$k$，设$n_k(d)$为最大度不超过$d$且直径不超过$k$的图的最大阶数。我们证明，对于每个固定的$k$，当$d\to\f$时，$\frac{n_k(d)}{d^k}=1$，从而解决了固定直径下的渐近度-直径问题，并证明了Bollobás的一个猜想。下界来自正则图$H_{k,q}$，其由素数幂$q$索引，顶点为$\f_q^{2k+1}$中的部分旗标，这些图的直径为$k$，阶数$|V(H_{k,q})|=(1+o(1))\triangle(H_{k,q})^k$。我们还为每个固定的$\f\be2$构造了最大度不超过$d$且线图直径不超过$\f$的图，其边数为$(1+o(1))d^{\f}$。
英文摘要
For positive integers $d$ and $k$, let $n_k(d)$ be the maximum order of a graph of maximum degree at most $d$ and diameter at most $k$. We prove that $$ \lim_{d\to\infty}\frac{n_k(d)}{d^k}=1$$ for every fixed $k$, thereby resolving the asymptotic degree-diameter problem for fixed diameter and proving a conjecture of Bollobás. The lower bound comes from regular graphs $H_{k,q}$, indexed by prime powers $q$, whose vertices are partial flags in $\mathbb{F}_q^{\,2k+1}$. These graphs have diameter $k$ and order $|V(H_{k,q})| =(1+o(1))Δ(H_{k,q})^k$. We also construct, for every fixed $\ell \ge 2$, graphs of maximum degree at most $d$ and line-graph diameter at most $\ell$ with $(1+o(1))d^{\ell}$ edges.
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