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arXiv · 2606.19742

Extremal eigenvalues of combinatorial Hodge Laplacians

Zhen Chen·Suil O·Jianfeng Wang
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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 $\lambda_{\max}(L_k^{\mathrm{up}})\le n$ in every dimension. We conjecture that $\lambda_{\max}(L_k^{\mathrm{up}})$ is in fact non-increasing in $k$, equivalently that $\sigma_{\max}(\partial_{k+1})\le\sigma_{\max}(\partial_k)$, 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 \[ \lambda_{\max}\big(L_k^{\mathrm{up}}\big)\ \le\ \nu_{k-1}+\tfrac1{k+2}\big(n-\nu_{k-1}\big), \qquad \nu_{k-1}=\lambda_{\max}\big(L_{k-1}^{\mathrm{up}}\big), \] 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, Stani\'c, Wang and Wang, and the first case reads $\alpha(G)\le2$. We also characterize the connected graphs of order at least seven with $\lambda_2(L_1)\le 3$ as the firefly graphs.

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Zhen Chen, Suil O, Jianfeng Wang. 2026-06-18. Extremal eigenvalues of combinatorial Hodge Laplacians. https://arxiv.org/abs/2606.19742

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