Majorization, degree sequence and [formula omitted]-spectral characterization of graphs

Abstract

Let π=(d1,d2,…,dn) and π′=(d1′,d2′,…,dn′) be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π≺π′, if and only if ∑i=1ndi=∑i=1ndi′ and ∑i=1jdi≤∑i=1jdi′ for j=1,2,…,n−1. By using majorization, we show that some graphs are determined by their Aα-spectra. Firstly, we show that the lollipop graph is determined by its Aα-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 Cámara and Haemers’ [Spectral characterizations of almost complete graphs, Discrete Appl. Math. 176 (2014) 19–23], we show Kn∖H is determined by its Aα-spectrum where H∈{Pk,K1,k,K2p+q∖pK2} and 1∕2<α<1, which implies that the friendship graph is determined by its Aα-spectrum for 1∕2<α<1.