Abstract: This paper studied lower bounds on the spectral radius of connected simple graphs and proved an useful inequality for the number of walks. Furthermore, some new lower bounds on the spectral radius of graphs were provided in terms of the maximum and minimum degree, the average degree, the 2-degree and the number of $k$-walks(with $k$ vertexes). By applying the properties of similar matrices and the Weyl inequalities,  another lower bound was obtained by means of the number of $k$-walks.  Simultaneously, all extremal graphs which achieve above bounds were also characterized.

Key words: adjacency matrix, spectral radius, Perron eigenvector, lower bound