<math xmlns='http://www.w3.org/1998/Math/MathML' id='M1'><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math>-Spectral Characterizations of Some Joins

LetGbe a graph withnvertices. For every realα∈0,1, writeAαGfor the matrixAαG=αDG+1−αAG, whereAGandDGdenote the adjacency matrix and the degree matrix ofG, respectively. The collection of eigenvalues ofAαGtogether with multiplicities are called theAα-spectrum ofG. A graphGis said to be determined by itsAα-spectrum if all graphs having the sameAα-spectrum asGare isomorphic toG. In this paper, we show that some joins are determined by theirAα-spectra forα∈0,1/2 or 1/2,1.