Two upper bounds on the A_α-spectral radius of a connected graph

If A(G) and D(G) are respectively the adjacency matrix and the diagonal matrix of vertex degrees of a connected graph G, the generalized adjacency matrix Aα(G) is defined as Aα(G)=α D(G)+(1-α) A(G), where 0≤ α ≤ 1. The Aα (or generalized) spectral radius λ(Aα(G)) (or simply λα) is the largest eigenvalue of Aα(G). In this paper, we show that                                           λα ≤αΔ+(1-α)(2m(1-1/ω))1/2, where m, Δ and ω=ω(G) are respectively the size, the largest degree and the clique number of $G$. Further, if G has order n, then we show that                      2λα ≤ max1≤i≤n [αdi  + √α2 di ^2 +4mi(1-α)[α+(1-α)mj]  where di  and mi  are respectively the degree and the average 2-degree of the vertex vi.