The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let $${\mathcal F}\left( \lambda \right)$$ ℱ ( λ ) be the family of connected graphs of spectral radius ≤ λ. We show that $${\mathcal F}\left( \lambda \right)$$ ℱ ( λ ) can be defined by a finite set of forbidden subgraphs if and only if $$\lambda > \lambda *: = \sqrt {2 + \sqrt 5 } \approx 2.058$$ λ < λ * : = 2 + 5 ≈ 2.058 and λ ∉ {α 2 , α 3 , …}, where $${\alpha _m} = \beta _m^{1/2} + \beta _m^{ - 1/2}$$ α m = β m 1 / 2 + β m − 1 / 2 and β m is the largest root of x m +1 = 1+ x + … + x m −1 . The study of forbidden subgraphs characterization for $${\mathcal F}\left( \lambda \right)$$ ℱ ( λ ) is motivated by the problem of estimating the maximum cardinality of equiangular lines in the n -dimensional Euclidean space ℝ n family of lines through the origin such that the angle between any pair of them is the same. Denote by N α ( n ) the maximum number of equiangular lines in ℝ n with angle arccos α . We establish the asymptotic formula N α ( n ) = c α n + O α (1) for every $${N_\alpha }\left( n \right) = {c_\alpha }n + {O_\alpha }\left( 1 \right)$$ N α ( n ) = c α n + O α ( 1 ) . In particular, $$\alpha \ge {1 \over {1 + 2\lambda *}}$$ α ≥ 1 1 + 2 λ * . Besides we show that $${N_{1/3}}\left( n \right) = 2n + O\left( 1 \right)\quad {\rm{and}}\quad {N_{1/5}}\left( n \right),\,{N_{1/(1 + 2\sqrt 2 )}}(n) = {3 \over 2}n\, + O\left( 1 \right).$$ N 1 / 3 ( n ) = 2 n + O ( 1 ) and N 1 / 5 ( n ) , N 1 / ( 1 + 2 2 ) ( n ) = 3 2 n + O ( 1 ) . , which improves a recent result of Balla, Dräxler, Keevash and Sudakov.

中文翻译：

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有界谱半径图的禁止子图，适用于等角线

图的谱半径是其邻接矩阵的最大特征值。令$${\mathcal F}\left( \lambda \right)$$ ℱ ( λ ) 是谱半径≤ λ 的连通图族。我们证明 $${\mathcal F}\left( \lambda \right)$$ ℱ ( λ ) 可以由一组有限的禁止子图定义当且仅当 $$\lambda > \lambda *: = \sqrt {2 + \sqrt 5 } \approx 2.058$$ λ < λ * : = 2 + 5 ≈ 2.058 和 λ ∉ {α 2 , α 3 , …}，其中 $${\alpha _m} = \beta _m^{ 1/2} + \beta _m^{ - 1/2}$$ α m = β m 1 / 2 + β m − 1 / 2 并且 β m 是 xm +1 = 1+ x + … + 的最大根xm -1 。$${\mathcal F}\left( \lambda \right)$$ ℱ ( λ ) 的禁止子图表征的研究是由估计 n 维欧几里得空间中等角线的最大基数的问题 ℝ n通过原点的线族，使得任何一对之间的角度都相同。用N α ( n ) 表示ℝ n 中角度为arccos α 的等角线的最大数目。我们为每一个$${N_\alpha }\left( n \right) = {c_\alpha }n + {O_\alpha }\ 建立渐近公式N α ( n ) = c α n + O α (1) left( 1 \right)$$ N α ( n ) = c α n + O α ( 1 ) 。特别地， $$\alpha \ge {1 \over {1 + 2\lambda *}}$$ α ≥ 1 1 + 2 λ * 。此外，我们证明 $${N_{1/3}}\left( n \right) = 2n + O\left( 1 \right)\quad {\rm{and}}\quad {N_{1/5} }\left( n \right),\,{N_{1/(1 + 2\sqrt 2 )}}(n) = {3 \over 2}n\, + O\left( 1 \right). $$ N 1 / 3 (n) = 2 n + O ( 1 ) 和 N 1 / 5 ( n ) , N 1 / ( 1 + 2 2 ) ( n ) = 3 2 n + O ( 1 ) 。，这改进了 Balla、Dräxler、Keevash 和 Sudakov 最近的结果。