On the eigenvalues and eigenvectors of Aα in weighted non-regular graphs
Let Gω=(G,ω)be a weighted graph,whose adjacency matrix and weighted degree diagnoal matrix are A(Gω)and D(Gω),respectively.For given α ∈[0,1],the matrix Aα(Gω)=αD(Gω)+(1-α)A(Gω)is the Aα-matrix of Gω.In this paper,we give some bounds on the Aα-eigenvalue of connected weighted non-regular graphs Gω,and obtain the lower bound of the ratio of the largest component to the smallest component in the eigenvector of the Aα-spectral radius.
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