Maxima of the $Q$-index for outer-planar graphs

The $Q$-$index$ of graph $G$ is the largest eigenvalue $q(G)$ of its signless Laplacian $Q(G)$. In this paper, we prove that the graph $K_1\mathrm{\nabla }{P}_{n-1}$ has the maximal $Q$-index among all outer-planar graphs of order $n$.