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 $alphainleft[ 0,1right] $, write $A_{alpha}left( Gright) $ for the matrix [ A_{alpha}left( Gright) =alpha Dleft( Gright) +(1-alpha)Aleft( Gright) . ] This paper presents some extremal results about the spectral radius $rho_{alpha}left( Gright) $ of $A_{alpha}left( Gright) $ that generalize previous results about $rho_{0}left( Gright) $ and $rho _{1/2}left( Gright) $. 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 $alphainleft[ 0,1right) $ and $G$ is a graph of order $n$ and diameter at least $k,$ then% [ rho_{alpha}(G)leqrho_{alpha}(B_{n-k+2,lfloor k/2rfloor,lceil k/2rceil}), ] with equality holding if and only if $G=B_{n-k+2,lfloor k/2rfloor,lceil k/2rceil}$. This result generalizes results of Hansen and Stevanovi{c} cite{HaSt08}, and Liu and Lu cite{LiLu14}.
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