On the sum of the two largest eigenvalues of the curl-curl operator on graphs

The Grone–Merris conjecture, proved by Bai in~2011, states that the spectrum of the graph Laplacian ${\mathrm{\Delta }}_0=-\mathrm{div}\mathrm{grad}$ 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 ${\mathrm{curl}}^{\ast }\mathrm{curl}$ 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 ${\mathrm{curl}}^{\ast }\mathrm{curl}$ 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 ${\mathrm{curl}}^{\ast }\mathrm{curl}$. As a corollary, we obtain upper bounds for the two largest eigenvalues of the full graph Helmholtzian ${\mathrm{\Delta }}_1=-\mathrm{grad}\mathrm{div}+{\mathrm{curl}}^{\ast }\mathrm{curl}$. The same result extends to the up-Laplacian of any $3$-family, yielding a concrete step towards the Duval–Reiner conjecture in dimension~$1$.