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Abstract:We establish operator-norm bounds for discrete Hodge Laplacians on weighted graphs and simplicial complexes; essential self-adjointness on natural cores follows as a corollary of these bounds. On unweighted $d$-regular graphs ($d\ge 2$) we prove, via a line-complex reduction, the effective bound $\|\widetilde{\Delta}_{1}\|\le \Delta(L(G))+2$ for the edge block. Combined with $\Delta(L(G))\le 2(d-1)$, it gives the announced $\|\widetilde{\Delta}_{1}\|\le 2d$, and hence the looser Schur-type bound $\|\widetilde{\Delta}_{1}\|\le 4(d-1)$. The first of these is sharp: it is attained on every unweighted $d$-regular \emph{bipartite} graph, finite or infinite. Weighted extensions are obtained via a comparability constant. No geometric completeness or curvature assumption is needed. Throughout, we work with flag (clique) complexes, i.e.\ simplicial complexes whose $k$-simplices are precisely the $(k+1)$-cliques of a weighted underlying graph. The method extends to higher degrees through dual up/down degrees. On any countable complex, an ordered coloring yields a unitarily equivalent reformulation of the skew model on \emph{unoriented} simplices, via canonical (color-ordered) representatives. On non-bipartite graphs the bound is strict. We determine the gap on standard periodic lattices by a Floquet--Bloch computation: the exact norms are $9$ on the triangular lattice, where $2d=12$, and $16$ on the face-centered cubic lattice, where $2d=24$.

Submission history

From: Jadlaoui Amel [view email]
[v1] Tue, 21 Oct 2025 14:11:29 UTC (20 KB)
[v2] Thu, 23 Oct 2025 15:06:00 UTC (21 KB)
[v3] Thu, 6 Aug 2026 12:14:02 UTC (44 KB)