Abstract:The Grone--Merris conjecture, proved by Bai in~2011, states that the spectrum of the graph Laplacian $\Delta_0 = -\operatorname{div}\operatorname{grad}$ is majorized by the conjugate of the vertex degree sequence. Duval and Reiner proposed a simplicial complex analogue of this statement. On a graph, where triangles serve as $2$-simplices, their conjecture reduces to the assertion that the spectrum of $\operatorname{curl}^*\operatorname{curl}$ is majorized by the conjugate of the second-order degree sequence, which records the number of triangles containing each vertex.
We prove that the sum of the two largest eigenvalues of $\operatorname{curl}^*\operatorname{curl}$ does not exceed the sum of the first two entries of that conjugate sequence. This confirms the first two majorization inequalities predicted by Duval and Reiner for $\operatorname{curl}^*\operatorname{curl}$. As a corollary, we obtain upper bounds for the two largest eigenvalues of the full graph Helmholtzian $\Delta_1 = -\operatorname{grad}\operatorname{div} + \operatorname{curl}^*\operatorname{curl}$. The same result extends to the up-Laplacian of any $3$-family, yielding a concrete step towards the Duval--Reiner conjecture in dimension~$1$.
From: Lu Lu [view email]
[v1]
Thu, 25 Jun 2026 01:36:22 UTC (661 KB)
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