Inspired by persistent homology in topological data analysis, we introduce the homological eigenvalues of the graph $p$-Laplacian ${\mathrm{\Delta }}_p$, which allows us to analyze and classify non-variational eigenvalues. We show the stability of homological eigenvalues, and we prove that for any homological eigenvalue $\lambda ({\mathrm{\Delta }}_p)$, the function $p\mapsto p{(2\lambda ({\mathrm{\Delta }}_p))}^{\frac{1}{p}}$ is locally increasing, while the function $p\mapsto 2^{-p}\lambda ({\mathrm{\Delta }}_p)$ is locally decreasing. As a special class of homological eigenvalues, the min–max eigenvalues ${\lambda }_1({\mathrm{\Delta }}_p),\dots ,{\lambda }_k({\mathrm{\Delta }}_p),\dots ,$ are locally Lipschitz continuous with respect to $p\in [1,+\infty )$. We also establish the monotonicity of $p{(2{\lambda }_k({\mathrm{\Delta }}_p))}^{\frac{1}{p}}$ and $2^{-p}{\lambda }_k({\mathrm{\Delta }}_p)$ with respect to $p\in [1,+\infty )$.

These results systematically establish a refined analysis of ${\mathrm{\Delta }}_p$-eigenvalues for varying $p$, which lead to several applications, including: (1) settle an open problem by Amghibech on the monotonicity of some function involving eigenvalues of $p$-Laplacian with respect to $p$; (2) resolve a question asking whether the third eigenvalue of graph $p$-Laplacian is of min–max form; (3) refine the higher-order Cheeger inequalities for graph $p$-Laplacians by Tudisco and Hein, and extend the multi-way Cheeger inequality by Lee, Oveis Gharan and Trevisan to the $p$-Laplacian case.

Furthermore, for the 1-Laplacian case, we characterize the homological eigenvalues and min–max eigenvalues from the perspective of topological combinatorics, where our idea is similar to the authors’ work on discrete Morse theory.

受拓扑数据分析中持久同源性的启发，我们引入了图的同调特征值$p$-拉普拉斯算子${\mathrm{\Delta }}_p$，这使我们能够对非变分特征值进行分析和分类。我们证明了同调特征值的稳定性，并且证明对于任何同调特征值$\lambda （{\mathrm{\Delta }}_p）$， 功能$p\mapsto p{（2\lambda （{\mathrm{\Delta }}_p））}^{\frac{1}{p}}$是局部递增的，而函数$p\mapsto 2^{-p}\lambda （{\mathrm{\Delta }}_p）$是局部递减的。作为一类特殊的同调特征值，最小-最大特征值${\lambda }_1（{\mathrm{\Delta }}_p）,\dots \dots ,{\lambda }_k（{\mathrm{\Delta }}_p）,\dots \dots ,$局部 Lipschitz 连续$p\epsilon [1,+\mathrm{无穷大}）$。我们还建立了单调性$p{（2{\lambda }_k（{\mathrm{\Delta }}_p））}^{\frac{1}{p}}$和$2^{-p}{\lambda }_k（{\mathrm{\Delta }}_p）$关于$p\epsilon [1,+\mathrm{无穷大}）$。

这些结果系统地建立了对${\mathrm{\Delta }}_p$- 变化的特征值$p$，这导致了几个应用，包括：（1）解决了 Amghibech 的一个开放问题，涉及涉及特征值的某些函数的单调性$p$- 拉普拉斯算子$p$; (2) 解决图的第三个特征值是否存在的问题$p$-拉普拉斯算子是最小–最大形式；(3) 细化图的高阶 Cheeger 不等式$p$- Tudisco 和 Hein 提出的拉普拉斯算子，并将 Lee、Oveis Gharan 和 Trevisan 提出的多路 Cheeger 不等式扩展到$p$-拉普拉斯案例。

此外，对于1-拉普拉斯情况，我们从拓扑组合学的角度描述了同调特征值和最小-最大特征值，我们的想法类似于作者在离散莫尔斯理论方面的工作。