Houdr\'e and Tetali defined a class of isoperimetric constants $\varphi_p$ of graphs for $0 \leq p \leq 1$, and conjectured a Cheeger-type inequality for $\varphi_\frac12$ of the form $$\lambda_2 \lesssim \varphi_\frac12 \lesssim \sqrt{\lambda_2}$$ where $\lambda_2$ is the second smallest eigenvalue of the normalized Laplacian matrix. If true, the conjecture would be a strengthening of the hard direction of the classical Cheeger's inequality. Morris and Peres proved Houdr\'e and Tetali's conjecture up to an additional log factor, using techniques from evolving sets. We present the following related results on this conjecture. - We provide a family of counterexamples to the conjecture of Houdr\'e and Tetali, showing that the logarithmic factor is needed. - We match Morris and Peres's bound using standard spectral arguments. - We prove that Houdr\'e and Tetali's conjecture is true for any constant $p$ strictly bigger than $\frac12$, which is also a strengthening of the hard direction of Cheeger's inequality. Furthermore, our results can be extended to directed graphs using Chung's definition of eigenvalues for directed graphs.

翻译：Houdré与Tetali为图定义了一类等周常数$\varphi_p$，其中$0 \leq p \leq 1$，并针对$\varphi_\frac12$提出了一个Cheeger型不等式的猜想，其形式为$$\lambda_2 \lesssim \varphi_\frac12 \lesssim \sqrt{\lambda_2}$$，其中$\lambda_2$是归一化拉普拉斯矩阵的第二小特征值。若该猜想成立，将是对经典Cheeger不等式困难方向的强化。Morris与Peres运用演化集方法，在附加一个对数因子的条件下证明了Houdré与Tetali的猜想。本文就该猜想提出以下相关结果：- 我们构造了一族反例，表明Houdré与Tetali的猜想需要引入对数因子。- 我们通过标准谱方法达到了Morris与Peres给出的界。- 我们证明了对于任意严格大于$\frac12$的常数$p$，Houdré与Tetali的猜想均成立，这同样是对Cheeger不等式困难方向的强化。此外，利用Chung针对有向图定义的特征值，我们的结果可推广至有向图情形。