  Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the
diagonal matrix of the degrees of $G$. For every real $\alpha\in\left[
0,1\right] $, write $A_{\alpha}\left( G\right) $ for the matrix \[
A_{\alpha}\left( G\right) =\alpha D\left( G\right) +(1-\alpha)A\left( G\right)
. \] This paper presents some extremal results about the spectral radius
$\rho_{\alpha}\left( G\right) $ of $A_{\alpha}\left( G\right) $ that generalize
previous results about $\rho_{0}\left( G\right) $ and $\rho _{1/2}\left(
G\right) $.
  In particular, write $B_{p,q,r}$ be the graph obtained from a complete graph
$K_{p}$ by deleting an edge and attaching paths $P_{q}$ and $P_{r}$ to its
ends. It is shown that if $\alpha\in\left[ 0,1\right) $ and $G$ is a graph of
order $n$ and diameter at least $k,$ then% \[
\rho_{\alpha}(G)\leq\rho_{\alpha}(B_{n-k+2,\lfloor k/2\rfloor,\lceil
k/2\rceil}), \] with equality holding if and only if $G=B_{n-k+2,\lfloor
k/2\rfloor,\lceil k/2\rceil}$. This result generalizes results of Hansen and
Stevanovi\'{c} \cite{HaSt08}, and Liu and Lu \cite{LiLu14}.
