  In 1990, Cvetkovi\'{c} and Rowlinson [The largest eigenvalue of a graph: a
survey, Linear Multilinear Algebra 28(1-2) (1990), 3--33] conjectured that
among all outerplanar graphs on $n$ vertices, $K_1\vee P_{n-1}$ attains the
maximum spectral radius. In 2017, Tait and Tobin [Three conjectures in extremal
spectral graph theory, J. Combin. Theory, Ser. B 126 (2017) 137-161] confirmed
the conjecture for sufficiently large values of $n$. In this article, we show
the conjecture is true for all $n\geq2$ except for $n=6$.
