有界顶点连通度的正则图的极值谱隙

中文摘要 AI 辅助

Fiedler 的著名不等式保证了连通非完全 $r$-正则图 $G$ 的谱隙 $r-\lambda_2(G)$ 由上界为 $G$ 的顶点连通度 $\kappa(G)$。我们证明，对于整数 $t\geq2$ 且 $r>2t^2$，每个顶点连通度至多为 $2t$ 的连通 $r$-正则图具有至多 $\frac{1}{2}(r+t+2-\sqrt{(r-t+2)^2-4t(t-1)})$ 的谱隙。我们表明，对于每个固定的 $t \ge 2$ 以及足够大的满足 $r+1$ 能被 $t$ 整除的 $r > 2t^2$，该界是近乎最优的。我们进一步证明，如果 $r>(2t-1)(2t-2)$，那么每个顶点连通度为奇数且不超过 $2t-1$ 的连通 $r$-正则图具有至多 $r-\min\bigl\{\xi(r,t),\nu(r,t)\bigr\}$ 的谱隙，其中 $\xi(r,t)$ 和 $\nu(r,t)$ 是某些明确给出的函数。如果 $r\geq6t^2$，则该最小值等于 $\xi(r,t)=\frac{r(2r-4t+5)}{2(r-t+2)}$。特别地，当 $\kappa(G)=2t-1$ 时，这给出 $r-\lambda_2(G)<t=(\kappa(G)+1)/2$，这比 Fiedler 的界改进了近两倍。我们的上界在其参数范围内改进了关于正则图谱隙的两个已知界。

英文摘要

The well-known inequality of Fiedler ensures that the spectral gap $r-λ_2(G)$ of a connected non-complete $r$-regular graph $G$ is bounded from above by the vertex connectivity $κ(G)$ of $G$. We prove that, for integers $t\geq2$ and $r>2t^2$, every connected $r$-regular graph with vertex connectivity at most $2t$ has spectral gap at most $\frac{1}{2}(r+t+2-\sqrt{(r-t+2)^2-4t(t-1)})$. We show that this bound is nearly optimal for each fixed $t \ge 2$ and sufficiently large $r > 2t^2$ such that $r+1$ is divisible by $t$. We further prove that if $r>(2t-1)(2t-2)$, then every connected $r$-regular graph whose vertex connectivity is odd and no more than $2t-1$ has spectral gap at most $r-\min\bigl\{ξ(r,t),ν(r,t)\bigr\}$ for some explicitly given functions $ξ(r,t)$ and $ν(r,t)$. If $r\geq6t^2$, then this minimum equals $ξ(r,t)=\frac{r(2r-4t+5)}{2(r-t+2)}$. In particular, when $κ(G)=2t-1$, this gives $r-λ_2(G)<t=(κ(G)+1)/2$, which improves Fiedler's bound by nearly a factor of two. Our upper bounds in their parameter ranges improve two known bounds on the spectral gap of regular graphs.

发表机构

• College of Mathematics and System Science, Xinjiang University（新疆大学数学与系统科学学院）
• School of Mathematics and Statistics, The University of Melbourne（墨尔本大学数学与统计学院）

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