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

Mathematics > Combinatorics [Submitted on 18 Jun 2026] Title:An upper bound on the largest eigenvalue of the Helmholtzian of a graph View PDF HTML (experimental)Abstract:The Helmholtzian of a graph $G$ is the Hodge $1$-Laplacian $L_1=L_1^{\mathrm{up}}+L_1^{\mathrm{down}}$ of its clique complex, built from the triangle--edge and edge--vertex boundary operators $\partial_2$ and $\partial_1$. 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 $\lambda_{\max}(L_1^{\mathrm{up}})\le\mu_1(G)$. We recast it as a question about the complement of $G$: localizing $L_1^{\mathrm{up}}$ on the cycle space of $K_n$ turns it into the inequality $\lambda_{\min}(\bar L|_{Z_1})\ge a(\overline{G})$, where $\bar L$ is the up Laplacian of the missing triangles of $G$ and $a(\overline{G})=n-\mu_1(G)$ is the algebraic connectivity of the complement. From this viewpoint, we prove the unconditional bound \[ \lambda_{\max}\!\big(L_1^{\mathrm{up}}(G)\big)\ \le\ \mu_1(G)+\frac13\big(n-\mu_1(G)\big), \] which refines the integrality ceiling $\lambda_{\max}(L_1^{\mathrm{up}})\le n$ 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 $\overline{G}$, 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. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.