On the Aα-spectral radius of a graph

Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For any real α∈[0,1], Nikiforov [3] defined the matrix Aα(G) asAα(G)=αD(G)+(1−α)A(G). The largest eigenvalue of Aα(G) is called the Aα-spectral radius of G. In this paper, we give three edge graft transformations on Aα-spectral radius. As applications, we determine the unique graph with maximum Aα-spectral radius among all connected graphs with diameter d, and determine the unique graph with minimum Aα-spectral radius among all connected graphs with given clique number. In addition, some bounds on the Aα-spectral radius are obtained.