Bounds for the spectral radius of the $$A_{\alpha }$$ A α -matrix of graphs

For a simple graph G, let A(G) be the adjacency matrix and D(G) be the diagonal matrix of the vertex degrees of graph G. For a real number $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] , the generalized adjacency matrix $$A_{\alpha }(G)$$ A α ( G ) is defined as $$A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$$ A α ( G ) = α D ( G ) + ( 1 - α ) A ( G ) . The largest eigenvalue of the matrix $$A_{\alpha }(G)$$ A α ( G ) is the generalized adjacency spectral radius or the $$A_{\alpha }$$ A α -spectral radius of the graph G. In this paper, we obtain some new sharp lower and upper bounds for the generalized adjacency spectral radius of G, in terms of different parameters like vertex degrees, the maximum and the second maximum degrees, the number of vertices and the number of edges, etc, associated with the structure of graph G. The extremal graphs attaining these bounds are characterized. We show that our bounds improve some recent given bounds in the literature in some cases. Further, our results extend some known results for the adjacency and/or the signless Laplacian spectral radius of a graph G to a general setting.