On the A_α-spectral radius of connected graphs^*

For a simple graph G, the generalized adjacency matrix A_α(G) is defined as A_α(G) = αD(G) + (1 − α)A(G), α ∈ [0, 1], where A(G) is the adjacency matrix and D(G) is the diagonal matrix of the vertex degrees. It is clear that A₀(G) = A(G) and 2A₂ 1 (G) = Q(G) implying that the matrix A_α(G) is a generalization of the adjacency matrix and the signless Laplacian matrix. In this paper, we obtain some new upper and lower bounds for the generalized adjacency spectral radius λ(A_α(G)), in terms of vertex degrees, average vertex 2-degrees, the order, the size, etc. The extremal graphs attaining these bounds are characterized. We will show that our bounds are better than some of the already known bounds for some classes of graphs. We derive a general upper bound for λ(A_α(G)), in terms of vertex degrees and positive real numbers b_i. As application, we obtain some new upper bounds for λ(A_α(G)). Further, we obtain some relations between clique number ω(G), independence number γ(G) and the generalized adjacency eigenvalues of a graph G.