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

Let G be a simple connected graph of order n with degree sequence (d1, d2, …, dn). Denote (αt)i = Σj: i∼jdjα, (αm)i = (αt)i/diα and (αN)i = Σj: i∼j (αt)j, where α is a real number. Denote by λ1(G) and µ1(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 λ1(G) and µ1(G) in terms of (αt)i, (αm)i and (αN)i. Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.