two upper bounds on the 𝐴α-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≤α δ+(1−α)2m(1−1ω)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 √λα d𝑖+ α^2 𝑑𝑖^ 2+4m𝑖(1α)[α+(1α)m𝑖 where 𝑑𝑖 and 𝑚𝑖 are respectively the degree and the average 2degree of the vertex 𝑣𝑖 .