Abstract
The Aα matrix of a graph G is defined by Aα(G)=αD(G)+(1−α)A(G),0≤α≤1, where D(G) is the diagonal matrix of degrees and A(G) is the adjacency matrix of G. The Aα-spectrum of a graph G, denoted by SpecAα(G), is the set of eigenvalues together with their multiplicities of Aα(G). A graph G is said to be determined by the generalized Aα-spectrum (DGAαS for short), if any graph H with SpecAα(G)=SpecAα(H) and SpecAα(G¯)=SpecAα(H¯), is isomorphic to G. In this paper, we present a simple arithmetic condition for an almost α-controllable graph being DGAαS, which generalizes the main results in [17].
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