二分图、随机图与林-陆-丘曲率
Bipartite graphs, random graphs, and Lin--Lu--Yau curvature
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中文总结 AI 辅助
研究二分图的林-陆-丘曲率，通过给出边数及最小度条件来判定二分图具有正曲率，界是紧的，还从概率角度放宽边密度条件，基于新公式证明，揭示相对稠密随机二分图的正曲率特性。
中文摘要 AI 辅助
设\(G=(X,Y;E)\)为一个二分图，其部分\(X\)和\(Y\)满足\(|X| = m\)且\(|Y| = n\)。我们证明，边数多于\(mn - D(m,n)\)的每个二分图具有正的林-陆-丘曲率，其中当\(n\geq2m\)时\(D(m,n)=m - 2+\lceil\frac{n}{2}\rceil\)，当\(m\leq n<2m\)时\(D(m,n)=n - 1\)。我们还证明，阶数为\(m + n\)且\(m\geq n\)、最小度至少为\(\min\{n,\lfloor\frac{m + n}{3}\rfloor + 1\}\)的每个二分图具有正的林-陆-丘曲率。两个界都是紧的。同时，概率上我们可以放宽上述结果中的边密度条件。结果表明相对稠密的随机二分图是正曲率的。我们所有的证明都基于一个二分图林-陆-丘曲率的新公式。
英文摘要
Let $G = (X, Y; E)$ be a bipartite graph with parts $X$ and $Y$ where $|X|=m$ and $|Y|=n$. We show that every bipartite graph with more than $mn - D(m,n)$ edges has positive Lin--Lu--Yau curvature, where $D(m,n)=m-2+\lceil{\frac {n}{2}\rceil} \text{ if $n\geq 2m$}, \mbox{and} \ n-1 \text{ if $m\leq n< 2m$}.$ We also show that every bipartite graph of order $m+n$ with $m \geq n$ and minimum degree at least $\min\{n, \lfloor{\frac{m+n}{3}\rfloor}+1\}$ has positive Lin--Lu--Yau curvature. Both bounds are sharp. Meanwhile probabilistically we can relax the edge density conditions in above results. It is shown that relatively dense random bipartite graph is positively curved. All of our proofs are based on a new formula for Lin--Lu--Yau curvature of bipartite graphs.
