A function on bounds of the spectral radius of graphs

Let G = (V, E) be a simple connected graph with n vertices. The degree of v(i) is an element of V and the average of degrees of the vertices adjacent to v(i) are denoted by d(i) and m(i), respectively. The spectral radius of G is denoted by rho(G). In this paper, we introduce a parameter into an equation of adjacency matrix, and obtain two inequalities for upper and lower bounds of spectral radius. By assigning different values to this parameter, one can obtain some new and existing results on spectral radius. Specially, if G is a nonregular graph, then rho(G) <= max(1 <= j<i <= n) {d(i)m(i) - d(j)m(j) + root(d(i)m(i) - d(j)m(j))(2) - 4d(i)d(j)(d(i) - d(j))(m(i) - m(j))/2(d(i) - d(j))}, and rho(G) <= max(1 <= j<i <= n) {d(i)m(i) - d(j)m(j) + root(d(i)m(i) - d(j)m(j))(2) - 4d(i)d(j)(d(i) - d(j))(m(i) - m(j))/2(d(i) - d(j))}. if G is a bidegreed graph whose vertices of same degree have equal average of degrees, then the equality holds.