  For $0\le \alpha\le 1$, Nikiforov proposed to study the spectral properties
of the family of matrices $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ of a graph
$G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency
matrix. The $\alpha$-spectral radius of $G$ is the largest eigenvalue of
$A_{\alpha}(G)$. We give upper bounds for $\alpha$-spectral radius for
unicyclic graphs $G$ with maximum degree $\Delta\ge 2$, connected irregular
graphs with given maximum degree and and some other graph parameters, and
graphs with given domination number, respectively. We determine the unique tree
with second maximum $\alpha$-spectral radius among trees, and the unique tree
with maximum $\alpha$-spectral radius among trees with given diameter. For a
graph with two pendant paths at a vertex or at two adjacent vertex, we prove
results concerning the behavior of the $\alpha$-spectral radius under
relocation of a pendant edge in a pendant path. We also determine the unique
graphs such that the difference between the maximum degree and the
$\alpha$-spectral radius is maximum among trees, unicyclic graphs and
non-bipartite graphs, respectively.
