On the $\alpha$-index of graphs with pendent paths

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 $$A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right)\mathrm{.}$$ 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 $\alpha \in [0,1)$ and $G$ is a graph of order $n$ and diameter at least $k,$ then% $${\rho }_{\alpha }(G)\le {\rho }_{\alpha }(B_{n-k+2,\lfloor k\mathrm{/}2\rfloor ,\lceil k\mathrm{/}2\rceil }),$$ 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}.