Bounds for the (Laplacian) spectral radius of graphs with parameter α

Let G be a simple connected graph of order n with degree sequence (d (1), d (2), aEuro broken vertical bar, d (n) ). Denote ( (alpha) t) (i) = I pound (j: i similar to j) d (j) (alpha) , ( (alpha) m) (i) = ( (alpha) t) (i) /d (i) (alpha) and ( (alpha) N) (i) = I pound (j: i similar to j) ( (alpha) t) (j) , where alpha is a real number. Denote by lambda(1)(G) and A mu(1)(G) the spectral radius of the adjacency matrix and the Laplacian matrix of G, respectively. In this paper, we present some upper and lower bounds of lambda(1)(G) and A mu(1)(G) in terms of ( (alpha) t) (i) , ( (alpha) m) (i) and ( (alpha) N) (i) . Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.