Let G be a simple graph with n vertices and m edges. To each graph G we can associate the following matrix \begin{eqnarray*}a_{ij}=\left\{\begin{array}{ll}\mathcal{F}(d_i,d_j),&\,i\sim j\\ 0,& otherwise, \end{array}\right.\end{eqnarray*}, where F is an appropriately chosen function with the property F(x,y)=F(y,x). Let PTI(G)(x)= xn+a1xn−1+a2xn−2+a3xn−3+· · ·+an be the characteristic polynomial of the matrix AF. In this paper, we consider the properties of the coefficients a1 and a2. Furthermore, we present some bounds of the F (di,dj)^2 by means of the eigenvalues of the matrix AF. Using these general results, we obtained some new bounds for the general Randić index R_1 of bipartite graphs.