Abstract Let π = ( d 1 , d 2 , … , d n ) and π ′ = ( d 1 ′ , d 2 ′ , … , d n ′ ) be two non-increasing degree sequences. We say π is majorizated by π ′ , denoted by π ≺ π ′ , if and only if ∑ i = 1 n d i = ∑ i = 1 n d i ′ and ∑ i = 1 j d i ≤ ∑ i = 1 j d i ′ for j = 1 , 2 , … , n − 1 . By using majorization, we show that some graphs are determined by their A α -spectra. Firstly, we show that the lollipop graph is determined by its A α -spectrum for 0 α 1 , which generalizes the result of Zhang, Liu, Zhang and Yong’s [The lollipop graph is determined by its Q -spectrum, Discrete Math. 309 (2009) 3364–3369]. Secondly, following the result of Camara and Haemers’ [Spectral characterizations of almost complete graphs, Discrete Appl. Math. 176 (2014) 19–23], we show K n ∖ H is determined by its A α -spectrum where H ∈ { P k , K 1 , k , K 2 p + q ∖ p K 2 } and 1 ∕ 2 α 1 , which implies that the friendship graph is determined by its A α -spectrum for 1 ∕ 2 α 1 .

中文翻译：

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图的专业化、度数序列和 Aα 谱表征

摘要 设π = ( d 1 , d 2 , … , dn ) 和 π ′ = ( d 1 ′ , d 2 ′ , … , dn ′ ) 为两个非递增度数序列。我们说 π 被 π ′ 主化，表示为 π ≺ π ′ ，当且仅当 ∑ i = 1 ndi = ∑ i = 1 ndi ′ 且 ∑ i = 1 jdi ≤ ∑ i = 1 jdi ′ for j = 1 ， 2 , … , n − 1 。通过使用专业化，我们表明一些图是由它们的 A α 谱决定的。首先，我们证明棒棒糖图是由它的 A α 谱决定的，对于 0 α 1 ，它概括了 Zhang、Liu、Zhang 和 Yong 的 [棒棒糖图是由它的 Q 谱决定的，离散数学。309 (2009) 3364–3369]。其次，根据 Camara 和 Haemers 的结果 [几乎完整图的光谱特征，离散应用程序。数学。176 (2014) 19-23]，我们表明 K n ∖ H 由其 A α 谱决定，其中 H ∈ { P k , K 1 , k ,