The Laplacian spectral radius of graphs on surfaces (Q2469524)

Let \(G\) be an \(n\)-vertex graph, \(n\geq 3\), with maximum degree \(\Delta\). The author gives upper bounds on the Laplacian spectral radius of \(G\) in the following cases: (a) \(G\) can be embedded on a surface of Euler genus \(\gamma\); (b) \(G\) is 4-connected and either the surface is the sphere or the embedding is 4-representative; (c) \(G\) is a maximal outerplanar graph; d) \(G\) is a Halin graph. All bounds are explicitly stated, but their common form is \(\Delta + O(\sqrt{n})\).