We prove that the maximum eigenvalue of the (both signed and unsigned) Laplacian of level $k$ Kikuchi graph of any graph $G$ with $m$ edges is at most $m+k$. This confirms four recent conjectures of Apte, Parekh, and Sud. As applications, we obtain that tensor products of one and two qubit product states achieve an approximation ratio of $5/8$ for Quantum Max Cut and $5/7$ for the XY Hamiltonian. Moreover, combining our bounds with the algorithms analyzed by Apte, Parekh, and Sud, yields efficient algorithms achieving an approximation ratio of $0.614$ for Quantum Max Cut and $0.674$ for the XY Hamiltonian. Finally, we also make modest progress on Brouwer's conjecture and improve Lew's bound on the sum of the top-$k$ eigenvalues of a Graph Laplacian.

翻译：我们证明，对于任意具有$m$条边的图$G$，其$k$级Kikuchi图的（有符号和无符号）拉普拉斯矩阵的最大特征值不超过$m+k$。这证实了Apte、Parekh和Sud近期提出的四个猜想。作为应用，我们得到：单量子比特和双量子比特乘积态的张量积在量子Max Cut问题中可实现$5/8$的近似比，在XY哈密顿量问题中可实现$5/7$的近似比。此外，将我们的界与Apte、Parekh和Sud分析过的算法相结合，可得到高效算法，在量子Max Cut问题中实现$0.614$的近似比，在XY哈密顿量问题中实现$0.674$的近似比。最后，我们在Brouwer猜想方面取得了适度进展，并改进了Lew关于图拉普拉斯矩阵前$k$大特征值之和的界。