The signless Laplacian spectral radius of bounded degree graphs on surfaces

Let $G$ be an $n$-vertex $(n \geq 3)$ simple graph embeddable on asurface of Euler genus $\gamma$ (the number of crosscaps plus twicethe number of handles). In this paper, we present upper bounds forthe signless Laplacian spectral radius of planar graphs, outerplanargraphs and Halin graphs, respectively, in terms of order and maximumdegree. We also demonstrate that our bounds are sometimes betterthan known ones. For outerplanar graphs without internal triangles,we determine the extremal graphs with the maximum and minimumsignless Laplacian spectral radii