Extremal eigenvalues of combinatorial Hodge Laplacians

Abstract

For a finite simplicial complex on $[n]$, the combinatorial Hodge Laplacian splits as $L_k=L_k^{\mathrm{up}}+L_k^{\mathrm{down}}$, and Duval and Reiner showed that $λ_{\max}(L_k^{\mathrm{up}})\le n$ in every dimension. We conjecture that $λ_{\max}(L_k^{\mathrm{up}})$ is in fact non-increasing in $k$, equivalently that $σ_{\max}(\partial_{k+1})\leσ_{\max}(\partial_k)$, and prove this unconditionally in two cases: when every missing $(k+1)$-face has at most $k+1$ missing facets, and for shifted complexes, where we also identify the extremal eigenvalue exactly, as the number of vertices lying in a $(k+1)$-face. In general we prove \[ λ_{\max}\big(L_k^{\mathrm{up}}\big)\ \le\ ν_{k-1}+\tfrac1{k+2}\big(n-ν_{k-1}\big), \qquad ν_{k-1}=λ_{\max}\big(L_{k-1}^{\mathrm{up}}\big), \] refining that ceiling. The proofs run through a localization on the cycle space $\ker\partial_k$, which turns the comparison into a statement about the complement. In dimension one the complex is the clique complex of a graph, $L_1$ is its Helmholtzian, the conjecture is a question of Lu, Shi, Stanić, Wang and Wang, and the first case reads $α(G)\le2$. We also characterize the connected graphs of order at least seven with $λ_2(L_1)\le 3$ as the firefly graphs.