图的谱和的确切最大值
The Exact Maximum of the Spectral Sum of Graphs
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中文总结 AI 辅助
研究了阶为\(n\geq5\)的简单图\(G\)的谱和\(S_2(G)\)，确定其确切最大值及等式情况。通过结合Ky Fan变分原理等方法，证明了相关猜想，加强了已有结果，涵盖了其他猜想，给出了最大化图及\(S_2(G)\)的上界等结论。
中文摘要 AI 辅助
对于阶为\(n\)的简单图\(G\)，令\(S_2(G)=\lambda_1(G)+\lambda_2(G)\)表示其谱和。我们确定了对于每个\(n\geq5\)，\(S_2(G)\)的确切最大值以及所有等式情况。同构意义下唯一的最大化图是适当平衡的完全二部图与孤立顶点的不相交并集的补图，其三部分的大小由\(n\)模\(7\)确定。记此图为\(K_n^\star\)，我们进一步表明\(S_2(K_n^\star)\leq\frac{8n}{7}-2\)，当且仅当\(7\mid n\)时取等号。这证明了Kumar、Liu、Monterde、Pragada和Tait的一个猜想，该猜想通过将Aouchiche - Hansen 2010猜想从连通图扩展到所有图并断言极值图的唯一性来加强了它。该结果还包含了Ebrahimi B.、Mohar、Nikiforov和Ahmady在2008年的猜想。证明结合了Ky Fan变分原理和加权Ferrers商的谱不等式，将问题简化为一个明确的族，其补图的关联秩为一。精确整数优化和单独的等式分析随后得出最大值和唯一性。
英文摘要
For a simple graph $G$ of order $n$, let $S_2(G)=λ_1(G)+λ_2(G)$ denote its spectral sum. We determine, for every $n\geq5$, the exact maximum of $S_2(G)$ and all equality cases. The unique maximizer, up to isomorphism, is the complement of the disjoint union of a suitably balanced complete bipartite graph and isolated vertices, with the sizes of its three parts determined by $n$ modulo $7$. Denoting this graph by $K_n^\star$, we further show that $ S_2(K_n^\star)\leq\frac{8n}{7}-2,$ with equality exactly when $7\mid n$. This proves a conjecture of Kumar, Liu, Monterde, Pragada and Tait, which strengthens the Aouchiche--Hansen 2010 conjecture by extending it from connected graphs to all graphs and by asserting uniqueness of the extremal graph. The result also subsumes the 2008 conjecture of Ebrahimi B., Mohar, Nikiforov, and Ahmady. The proof combines Ky Fan's variational principle with a spectral inequality for weighted Ferrers quotients to reduce the problem to an explicit family whose complements have incidence rank one. Exact integer optimization and a separate equality analysis then yield the maximum and uniqueness.
