Abstract To determine the Turan numbers of even cycles is a central problem of extremal graph theory. Even for C 6 , the Turan number is still open. Till now, the best known upper bound is given by Furedi, Naor and Verstraete [On the Turan number for the hexagon, Advances in Math.]. In 2010, Nikiforov posed a spectral version of extremal graph theory problem: what is the maximum spectral radius ρ of an H -free graph of order n ? Let e x s p ( n , H ) = max { ρ ( G ) | | V ( G ) | = n , H ⊈ G } . In contrast to the unsolved problem of Turan number of C 6 , we obtain the exact value of e x s p ( n , C 6 ) and characterize the unique extremal graph. The result also confirms Nikiforov’s conjecture [The spectral radius of graphs without paths and cycles of specified length, Linear Algebra Appl.] for k = 2 .

中文翻译：

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图的谱极值：禁六边形

摘要 确定偶数圈的图兰数是极值图论的一个核心问题。即使对于 C 6 ，图兰数仍然是开放的。到目前为止，最著名的上限是由 Furedi、Naor 和 Verstraete 给出的 [关于六边形的图兰数，数学进展]。2010 年，Nikiforov 提出了一个极值图论问题的谱版本：n 阶无 H 图的最大谱半径 ρ 是多少？让 exsp ( n , H ) = max { ρ ( G ) | | V ( G ) | = n , H ⊈ G } 。与未解决的 C 6 的图兰数问题相反，我们获得了 exsp ( n , C 6 ) 的精确值并刻画了唯一的极值图。结果还证实了 Nikiforov 的猜想 [没有指定长度的路径和循环的图的谱半径，线性代数应用] k = 2 。