An upper bound on the largest eigenvalue of the Helmholtzian of a graph

The Helmholtzian of a graph $G$ is the Hodge $1$-Laplacian $L_1=L_1^{\mathrm{u}\mathrm{p}}+L_1^{\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}}$ of its clique complex, built from the triangle–edge and edge–vertex boundary operators ${\mathrm{\partial }}_2$ and ${\mathrm{\partial }}_1$. Problem~5.5 of Lu, Shi, Stanić, Wang and Wang asks whether ${\lambda }_{\max }(L_1)={\mu }_1(G)$ for every graph $G$, where ${\mu }_1(G)$ is the largest Laplacian eigenvalue; by the Hodge decomposition this is equivalent to ${\lambda }_{\max }(L_1^{\mathrm{u}\mathrm{p}})\le {\mu }_1(G)$. We recast it as a question about the complement of $G$: localizing $L_1^{\mathrm{u}\mathrm{p}}$ on the cycle space of $K_n$ turns it into the inequality ${\lambda }_{\min }(\bar{L}{\mathrm{\mid }}_{Z_1})\ge a(\bar{G})$, where $\bar{L}$ is the up Laplacian of the missing triangles of $G$ and $a(\bar{G})=n-{\mu }_1(G)$ is the algebraic connectivity of the complement. From this viewpoint, we prove the unconditional bound $${\lambda }_{\max }\text{\hspace{0.17em}⁣}(L_1^{\mathrm{u}\mathrm{p}}(G))\le {\mu }_1(G)+\frac{1}{3}(n-{\mu }_1(G)),$$ which refines the integrality ceiling ${\lambda }_{\max }(L_1^{\mathrm{u}\mathrm{p}})\le n$ of Duval and Reiner and is sharp exactly when that ceiling is attained. We then isolate the single sharp inequality, on the dense part of $\bar{G}$, that stops the method short of Problem~5.5, and we show that the localization, the bound, and this obstruction all persist for the up Laplacian of an arbitrary finite simplicial complex, in every dimension.