Hodge Laplacians on Weighted Simplicial Complexes: Forms, Closures, and Bounded Realizations

Abstract

We establish operator-norm bounds for discrete Hodge Laplacians on weighted flag complexes of a fixed dimension $n$, whose $k$-simplices are the $(k+1)$-cliques of a weighted graph; essential self-adjointness on natural cores follows, with no completeness or curvature assumption. Dual up/down degrees give Schur-type bounds in every degree. At top degree a unitary conjugation turns ${\Delta }_n$ into an adjacency operator on the line-complex plus a diagonal potential, to which Schur's test applies directly. At $n=1$ this is the Anderson--Morley edge-degree bound $\Vert {\Delta }_1\Vert \le \max \{°u+°v\}$, obtained here on infinite and on weighted graphs, whence $\Vert {\Delta }_1\Vert \le 2d$ in the $d$-regular case. The sharpness analysis is carried out at $n=1$, for the edge block; in higher degrees we prove boundedness and essential self-adjointness, not sharpness. We give a spectral criterion for the attainment of $2d$, together with sufficient conditions: it is attained on every finite $d$-regular bipartite graph, and on every infinite one that is amenable. Amenability cannot be dropped: the $d$-regular tree, $d\ge 3$, has exact norm $d+2\sqrt{d-1}<2d$. The standard periodic lattices being amenable, we compute their exact norms from the Bloch symbols: $2d$ in the bipartite cases, against $9$ for the triangular and $16$ for the face-centered cubic lattice, on which it is strict. Finally, an ordered coloring which any countable complex admits reformulates the skew model unitarily on unoriented simplices, the orientation signs becoming a function of the colors alone.