6 problems

Higher-order Cheeger inequalities conjecture. There should hold related higher-order Cheeger inequalities for weighted graphs.

Let $h_k$ be the $k$-way Cheeger constant of a graph and let $\hat{h}_k$ denote its modified $k$-way Cheeger constant. Higher-order modified Cheeger constant conjecture. There exis…

Let $S_d$ be a $d$-dimensional simplicial complex, with upper and lower Cheeger constants $h(S_d)$ and $h_{down}(S_d)$, upper and lower Laplacians $\Delta_d^{up}$ and…

Let $K=(V_K,E_K)$ and $L=(V_L,E_L)$ be graphs with $K\hookrightarrow L$. The strong persistent Cheeger inequality conjectures that … where $h^{K,L}_s$ is the strong persistent Chee…

Let $\mathcal G$ be a connected quantum graph obtained from a graph by identifying each edge $\mathsf e$ with an interval $(0,\ell_{\mathsf e})$. Let $h(\mathcal G)$ be its Cheeger…

Higher-order spectral partitioning conjecture. There are $k$ eigenvalues close to zero if and only if the vertex set $V$ can be partitioned into $k$ subsets, each defining a sparse…