Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For any real α∈0,1, Nikiforov (2017) 7 defined the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G). Let u and v be two vertices of a connected graph G. Suppose that u and v are connected by a path w0(=v)w1⋯ws−1ws(=u) where d(wi)=2 for 1≤i≤s−1. Let Gp,s,q(u,v) be the graph obtained by attaching the paths Pp to u and Pq to v. Let s=0,1. Nikiforov and Rojo (2018) 9 conjectured that ρα(Gp,s,q(u,v))<ρα(Gp−1,s,q+1(u,v)) if p≥q+2. In this paper, we confirm the conjecture. As applications, firstly, the extremal graph with maximal Aα-spectral radius with fixed order and cut vertices is characterized. Secondly, we characterize the extremal tree which attains the maximal Aα-spectral radius with fixed order and matching number. These results generalize some known results.