论文标题
请注意无三角形图的最小和最大特征值的总和
Note on the sum of the smallest and largest eigenvalues of a triangle-free graph
论文作者
论文摘要
令$ g $为$ n $顶点的无三角形图，带有邻接矩阵eigenvalues $μ_1（g）\ geqμ_2（g）\ geq \ dots \ geqμ_n（g）$。在本文中，我们研究数量$$μ_1（g）+μ_n（g）。$$我们证明，对于任何无三角形的图$ g $，我们都有$$μ_1（g）+μ_n（g）\ leq（g）\ leq（3-2 \ sqrt {2}} {2}）n。$$对于常规图表，我们的常规图是定期的。我们还证明，在无三角形的强烈规则图中，higman-sims图达到了$$ \ frac {μ_1（g）+μ_n（g）} {n} {$$的最大值。
Let $G$ be a triangle-free graph on $n$ vertices with adjacency matrix eigenvalues $μ_1(G)\geq μ_2(G)\geq \dots \geq μ_n(G)$. In this paper we study the quantity $$μ_1(G)+μ_n(G).$$ We prove that for any triangle-free graph $G$ we have $$μ_1(G)+μ_n(G)\leq (3-2\sqrt{2})n.$$ This was proved for regular graphs by Brandt, we show that the condition on regularity is not necessary. We also prove that among triangle-free strongly regular graphs the Higman-Sims graph achieves the maximum of $$\frac{μ_1(G)+μ_n(G)}{n}.$$
